Series and Sequences (Convergence, Divergence)

In calculus and mathematical analysis, sequences and series are foundational concepts used to understand patterns, limits, and infinite sums. A sequence is a list of numbers in a specific order, while a series is the sum of the terms in a sequence.

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1. Sequences

A sequence is an ordered list of numbers. Each number in the list is called a term of the sequence, and sequences can be either finite or infinite. Sequences are often represented by:

$a_n=a_1,a_2,a_3,\dots$

Where $a_n$ represents the general term, or the nth term, of the sequence.

Example of a Sequence

The sequence of natural numbers:

$1,2,3,4,5,\dots$

Another example is the sequence of squares:

$1^2,2^2,3^2,\dots =1,4,9,16,25,\dots$

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2. Convergence of Sequences

A sequence is said to converge if the terms of the sequence approach a single, finite limit as $n$ approaches infinity. Mathematically, a sequence $a_n$ converges to a limit $L$ if:

$\underset{n\to \mathrm{\infty }}{lim}{a}_n=L$

If a sequence does not converge to a finite limit, it is said to diverge.

Example 1: Convergent Sequence

Problem:
Consider the sequence $a_n=\frac{1}{n}$. Determine whether the sequence converges or diverges.

Answer:

Step 1: Given Data:
The sequence is $a_n=\frac{1}{n}$.

Step 2: Solution:
Take the limit as $n$ approaches infinity:

$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}=0$

Since the limit exists and is finite, the sequence converges.

Step 3: Final Answer:
The sequence $a_n=\frac{1}{n}$ converges to $0$.

Example 2: Divergent Sequence

Problem:
Consider the sequence $a_n=n$. Determine whether the sequence converges or diverges.

Answer:

Step 1: Given Data:
The sequence is $a_n=n$.

Step 2: Solution:
Take the limit as $n$ approaches infinity:

$\underset{n\to \mathrm{\infty }}{lim}n=\mathrm{\infty }$

Since the limit is infinite, the sequence diverges.

Step 3: Final Answer:
The sequence $a_n=n$ diverges.

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3. Series

A series is the sum of the terms of a sequence. If $a_n$ is a sequence, the series formed by summing its terms is written as:

$S=a_1+a_2+a_3+\cdots =\sum _{n=1}^{\mathrm{\infty }}{a}_n$

• A series can either converge to a finite value or diverge to infinity or fail to settle on any value.

A series converges if the sum of its terms approaches a finite value as more and more terms are added.

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4. Convergence and Divergence of Series

• A series $\sum _{n=1}^{\mathrm{\infty }}{a}_n$ converges if the partial sums $S_N=a_1+a_2+\cdots +a_N$ approach a finite value as $N\to \mathrm{\infty }$.
• A series diverges if the partial sums do not approach a finite value.

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Common Tests for Convergence

1. Geometric Series:
   A geometric series is of the form:

$\sum _{n=0}^{\mathrm{\infty }}a{r}^n$

• If $|r|<1$, the series converges to:

$S=\frac{a}{1–r}$

• If $|r|\ge 1$, the series diverges.

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Example 3: Geometric Series

Problem:
Determine whether the series $\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{2}\right)}^n$ converges or diverges.

Answer:

Step 1: Given Data:
The series is geometric with $a=1$ and $r=\frac{1}{2}$.

Step 2: Solution:
Since $|r|=\frac{1}{2}<1$, the series converges.

Now, apply the formula for the sum of a geometric series:

$S=\frac{1}{1–\frac{1}{2}}=\frac{1}{\frac{1}{2}}=2$

Step 3: Final Answer:
The series converges to $2$.

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2. p-Series Test:
   A p-series is of the form:

$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^p}$

• If $p>1$, the series converges.
• If $p\le 1$, the series diverges.

Example 4: p-Series

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$ with $p=2$.

Step 2: Solution:
Since $p=2>1$, the series converges.

Step 3: Final Answer:
The series converges.

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3. Harmonic Series:
   The harmonic series is a special case of a p-series with $p=1$:

$\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$

• The harmonic series diverges.

Example 5: Harmonic Series

Problem:
Determine whether the harmonic series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$.

Step 2: Solution:
Since this is a harmonic series, it diverges.

Step 3: Final Answer:
The series diverges.

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5. Series Representation

Series are useful for representing complex functions. For example, the Taylor series and Maclaurin series represent functions as an infinite sum of terms involving powers of $x$. This allows us to approximate functions in a variety of contexts.

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6. Alternating Series and the Alternating Series Test

An alternating series is a series in which the signs of the terms alternate between positive and negative. The Alternating Series Test states that an alternating series $\sum (-1{)}^n{a}_n$ converges if the terms $a_n$ are decreasing and the limit of $a_n$ approaches zero as $n\to \mathrm{\infty }$.

Example 6: Alternating Series

Problem:
Determine whether the alternating series $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n}$.

Step 2: Solution:
Since the terms $\frac{1}{n}$ are decreasing and $\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}=0$, the series converges by the Alternating Series Test.

Step 3: Final Answer:
The series converges.

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7. Conclusion

Sequences and series form the building blocks of calculus and mathematical analysis. Understanding whether a sequence or series converges or diverges is essential in a variety of applications, from approximating functions to solving complex problems. By applying tests such as the geometric series test, p-series test, or alternating series test, we can determine the behavior of many important sequences and series.

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Question And Answer Library

Example 1: Convergent Sequence

Problem:
Determine whether the sequence $a_n=\frac{1}{n}$ converges or diverges.

Answer:

Step 1: Given Data:
The sequence is $a_n=\frac{1}{n}$.

Step 2: Solution:
Evaluate the limit as $n$ approaches infinity:
$\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}=0$.

Since the limit exists and is finite, the sequence converges.

Step 3: Final Answer:
The sequence converges to $0$.

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Example 2: Divergent Sequence

Problem:
Determine whether the sequence $a_n=n^2$ converges or diverges.

Answer:

Step 1: Given Data:
The sequence is $a_n=n^2$.

Step 2: Solution:
Evaluate the limit as $n$ approaches infinity:
$\underset{n\to \mathrm{\infty }}{lim}{n}^2=\mathrm{\infty }$.

Since the limit is infinite, the sequence diverges.

Step 3: Final Answer:
The sequence diverges.

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Example 3: Convergent Series

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$.

Step 2: Solution:
This is a $p$-series with $p=2$.
Since $p>1$, the series converges.

Step 3: Final Answer:
The series converges.

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Example 4: Divergent Series

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$.

Step 2: Solution:
This is a harmonic series, which diverges.
Thus, the series diverges.

Step 3: Final Answer:
The series diverges.

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Example 5: Geometric Series

Problem:
Determine whether the series $\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{3}\right)}^n$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=0}^{\mathrm{\infty }}{\left(\frac{1}{3}\right)}^n$.

Step 2: Solution:
This is a geometric series with $a=1$ and $r=\frac{1}{3}$.
Since $|r|<1$, the series converges.

Now, apply the formula for the sum of a geometric series:
$S=\frac{a}{1–r}=\frac{1}{1–\frac{1}{3}}=\frac{1}{\frac{2}{3}}=\frac{3}{2}$.

Step 3: Final Answer:
The series converges to $\frac{3}{2}$.

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Example 6: Convergence of an Alternating Series

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n}$.

Step 2: Solution:
Check if the terms $\frac{1}{n}$ are decreasing and if $\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}=0$.
Since $\frac{1}{n}$ is decreasing and $\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}=0$,
the series converges by the Alternating Series Test.

Step 3: Final Answer:
The series converges.

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Example 7: Divergence Test

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{2^n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{2^n}$.

Step 2: Solution:
This is a geometric series with $a=\frac{1}{2}$ and $r=\frac{1}{2}$.
Since $|r|<1$, the series converges.

Now, apply the formula for the sum of a geometric series:
$S=\frac{a}{1–r}=\frac{\frac{1}{2}}{1–\frac{1}{2}}=\frac{\frac{1}{2}}{\frac{1}{2}}=1$.

Step 3: Final Answer:
The series converges to $1$.

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Example 8: p-Series

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3}$ with $p=3$.

Step 2: Solution:
Since $p=3>1$, the series converges.

Step 3: Final Answer:
The series converges.

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Example 9: Ratio Test

Problem:
Use the ratio test to determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{n!}{n^n}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{n!}{n^n}$.

Step 2: Solution:
Compute the ratio:
$L=\underset{n\to \mathrm{\infty }}{lim}\left|\frac{a_{n+1}}{a_n}\right|=\underset{n\to \mathrm{\infty }}{lim}|\frac{(n+1)!}{(n+1{)}^{n+1}}\cdot \frac{n^n}{n!}|$.

This simplifies to:
$=\underset{n\to \mathrm{\infty }}{lim}\frac{(n+1)n^n}{(n+1{)}^{n+1}}$.

Evaluating gives:
$=\underset{n\to \mathrm{\infty }}{lim}\frac{(n+1)}{(n+1)(1+\frac{1}{n}{)}^n}=0$.

Since $L<1$, the series converges by the ratio test.

Step 3: Final Answer:
The series converges.

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Example 10: Comparison Test

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^4}$ converges or diverges using the comparison test.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^4}$.

Step 2: Solution:
Compare with the $p$-series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^p}$ with $p=4$.
Since $p=4>1$, the comparison series converges.

By the comparison test, since $\frac{1}{n^4}\le \frac{1}{n^p}$, the original series converges.

Step 3: Final Answer:
The series converges.

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Example 11: Limit Comparison Test

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2+n}$ converges or diverges using the limit comparison test.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2+n}$.

Step 2: Solution:
We compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$.
Calculate the limit:
$L=\underset{n\to \mathrm{\infty }}{lim}\frac{\frac{1}{n^2+n}}{\frac{1}{n^2}}=\underset{n\to \mathrm{\infty }}{lim}\frac{n^2}{n^2+n}=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{1+\frac{1}{n}}=1$.

Since $L$ is finite and positive, the series converges.

Step 3: Final Answer:
The series converges.

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Example 12: Alternating Series Test

Problem:
Determine whether the alternating series $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n^2}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n^2}$.

Step 2: Solution:
Check if $a_n=\frac{1}{n^2}$ is decreasing and if $\underset{n\to \mathrm{\infty }}{lim}{a}_n=0$.
Since $\frac{1}{n^2}$ is decreasing and $\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^2}=0$,
the series converges by the Alternating Series Test.

Step 3: Final Answer:
The series converges.

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Example 13: Divergent Harmonic Series

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$.

Step 2: Solution:
This is a harmonic series, which is known to diverge.

Step 3: Final Answer:
The series diverges.

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Example 14: Ratio Test for Factorial Series

Problem:
Use the ratio test to determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{n!}{n^n}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{n!}{n^n}$.

Step 2: Solution:
Compute the ratio:
$L=\underset{n\to \mathrm{\infty }}{lim}\left|\frac{a_{n+1}}{a_n}\right|=\underset{n\to \mathrm{\infty }}{lim}\frac{(n+1)!}{(n+1{)}^{n+1}}\cdot \frac{n^n}{n!}$.

Simplifying gives:
$=\underset{n\to \mathrm{\infty }}{lim}\frac{(n+1)n^n}{(n+1{)}^{n+1}}=\underset{n\to \mathrm{\infty }}{lim}\frac{n+1}{(n+1)(1+\frac{1}{n}{)}^n}=0$.

Since $L<1$, the series converges by the ratio test.

Step 3: Final Answer:
The series converges.

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Example 15: Convergence of Exponential Series

Problem:
Determine whether the series $\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}$ converges for all real $x$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}$.

Step 2: Solution:
Use the ratio test:
$L=\underset{n\to \mathrm{\infty }}{lim}\left|\frac{a_{n+1}}{a_n}\right|=\underset{n\to \mathrm{\infty }}{lim}\frac{x^{n+1}}{(n+1)!}\cdot \frac{n!}{x^n}=\underset{n\to \mathrm{\infty }}{lim}\frac{|x|}{n+1}=0$.

Since $L<1$, the series converges for all real $x$.

Step 3: Final Answer:
The series converges for all real $x$.

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Example 16: Convergence of Logarithmic Series

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{\ln (n)}{n^2}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{\ln (n)}{n^2}$.

Step 2: Solution:
Compare with the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^{1.5}}$ (using the limit comparison test).
Since $\ln (n)$ grows slower than any polynomial, the series converges.

Step 3: Final Answer:
The series converges.

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Example 17: Divergence of Series with Terms Increasing

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}n$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}n$.

Step 2: Solution:
The partial sums $S_N=1+2+\cdots +N=\frac{N(N+1)}{2}$ diverge to infinity.

Step 3: Final Answer:
The series diverges.

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Example 18: Alternating Series with Limit Zero

Problem:
Determine whether the alternating series $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n^3}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n^3}$.

Step 2: Solution:
Check if the terms $\frac{1}{n^3}$ are decreasing and if $\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^3}=0$.
Both conditions are satisfied, so the series converges by the Alternating Series Test.

Step 3: Final Answer:
The series converges.

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Example 19: Limit Comparison Test with Divergence

Problem:
Use the limit comparison test to determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2+1}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2+1}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$.
Calculate the limit:
$L=\underset{n\to \mathrm{\infty }}{lim}\frac{\frac{1}{n^2+1}}{\frac{1}{n^2}}=\underset{n\to \mathrm{\infty }}{lim}\frac{n^2}{n^2+1}=1$.

Since $L$ is finite and positive, both series converge or diverge together.
Since $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$ converges, so does $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2+1}$.

Step 3: Final Answer:
The series converges.

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Example 20: Ratio Test for Factorial Series

Problem:
Determine the convergence of the series $\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}$ using the ratio test.

Answer:

Step 1: Given Data:
The series is $\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}$.

Step 2: Solution:
Compute the ratio:
$L=\underset{n\to \mathrm{\infty }}{lim}\left|\frac{a_{n+1}}{a_n}\right|=\underset{n\to \mathrm{\infty }}{lim}\frac{x^{n+1}}{(n+1)!}\cdot \frac{n!}{x^n}=\underset{n\to \mathrm{\infty }}{lim}\frac{|x|}{n+1}=0$.

Since $L<1$ for all $x$, the series converges.

Step 3: Final Answer:
The series converges for all $x$.

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Example 21: Finding the Limit of a Sequence

Problem:
Find the limit of the sequence $a_n=\frac{3n+1}{2n+5}$ as $n\to \mathrm{\infty }$.

Answer:

Step 1: Given Data:
The sequence is $a_n=\frac{3n+1}{2n+5}$.

Step 2: Solution:
Evaluate the limit:
$\underset{n\to \mathrm{\infty }}{lim}\frac{3n+1}{2n+5}=\underset{n\to \mathrm{\infty }}{lim}\frac{3+\frac{1}{n}}{2+\frac{5}{n}}=\frac{3}{2}$.

Step 3: Final Answer:
The limit is $\frac{3}{2}$.

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Example 22: Finding the Sum of an Infinite Series

Problem:
Calculate the sum of the series $\sum _{n=0}^{\mathrm{\infty }}\frac{1}{3^n}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=0}^{\mathrm{\infty }}\frac{1}{3^n}$.

Step 2: Solution:
This is a geometric series with $a=1$ and $r=\frac{1}{3}$.
Since $|r|<1$, the series converges.

Now apply the formula for the sum:
$S=\frac{a}{1–r}=\frac{1}{1–\frac{1}{3}}=\frac{1}{\frac{2}{3}}=\frac{3}{2}$.

Step 3: Final Answer:
The sum is $\frac{3}{2}$.

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Example 23: Convergence of Alternating Series

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n}$.

Step 2: Solution:
Check if the terms $a_n=\frac{1}{n}$ are decreasing and if $\underset{n\to \mathrm{\infty }}{lim}{a}_n=0$.
Both conditions are satisfied, so the series converges by the Alternating Series Test.

Step 3: Final Answer:
The series converges.

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Example 24: Use of p-Series Test

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^{0.5}}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^{0.5}}$ with $p=0.5$.

Step 2: Solution:
Since $p=0.5\le 1$, the series diverges.

Step 3: Final Answer:
The series diverges.

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Example 25: Convergence of a Series with Logarithm

Problem:
Determine whether the series $\sum _{n=2}^{\mathrm{\infty }}\frac{\ln (n)}{n^2}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=2}^{\mathrm{\infty }}\frac{\ln (n)}{n^2}$.

Step 2: Solution:
Compare with the series $\sum _{n=2}^{\mathrm{\infty }}\frac{1}{n^{1.5}}$ (using the limit comparison test).
Since $\ln (n)$ grows slower than $n^{0.5}$, the series converges.

Step 3: Final Answer:
The series converges.

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Example 26: Divergence of a Series with Exponential Terms

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}{e}^{-n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}{e}^{-n}$.

Step 2: Solution:
This is a geometric series with $a=e^{-1}$ and $r=e^{-1}$.
Since $|r|<1$, the series converges.

Now apply the formula for the sum:
$S=\frac{e^{-1}}{1–e^{-1}}=\frac{e^{-1}}{1–\frac{1}{e}}=\frac{1}{e(1–\frac{1}{e})}=\frac{1}{e–1}$.

Step 3: Final Answer:
The series converges to $\frac{1}{e–1}$.

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Example 27: Telescoping Series

Problem:
Determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\left(\frac{1}{n}–\frac{1}{n+1}\right)$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\left(\frac{1}{n}–\frac{1}{n+1}\right)$.

Step 2: Solution:
This series telescopes:
$S_N=\left(\frac{1}{1}–\frac{1}{2}\right)+\left(\frac{1}{2}–\frac{1}{3}\right)+\dots +\left(\frac{1}{N}–\frac{1}{N+1}\right)$.

The terms simplify to:
$S_N=1–\frac{1}{N+1}$.

As $N\to \mathrm{\infty }$, $S_N\to 1$.
Thus, the series converges.

Step 3: Final Answer:
The series converges to $1$.

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Example 28: Comparison Test with Divergent Series

Problem:
Use the comparison test to determine if the series $\sum _{n=1}^{\mathrm{\infty }}\frac{2}{n+1}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{2}{n+1}$.

Step 2: Solution:
Compare with the harmonic series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$.
Since $\frac{2}{n+1}\sim \frac{2}{n}$ as $n\to \mathrm{\infty }$ and the harmonic series diverges, the original series diverges.

Step 3: Final Answer:
The series diverges.

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Example 29: Absolute Convergence

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n^2}$ converges absolutely or conditionally.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n^2}$.

Step 2: Solution:
Check absolute convergence by evaluating $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$.
Since $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$ converges, the original series converges absolutely.

Step 3: Final Answer:
The series converges absolutely.

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Example 30: Root Test

Problem:
Use the root test to determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^n}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^n}$.

Step 2: Solution:
Calculate the limit:
$L=\underset{n\to \mathrm{\infty }}{lim}\sqrt[n]{\frac{1}{n^n}}=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}=0$.

Since $L<1$, the series converges by the root test.

Step 3: Final Answer:
The series converges.

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Example 31: Divergence of Oscillating Series

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n}$.

Step 2: Solution:
This is an alternating series. Check if the terms $\frac{1}{n}$ are decreasing and if $\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}=0$.
Both conditions are satisfied, so the series converges by the Alternating Series Test.

Step 3: Final Answer:
The series converges.

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Example 32: Convergence of Series with Exponential Terms

Problem:
Determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{n^2}{2^n}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{n^2}{2^n}$.

Step 2: Solution:
Use the ratio test:
$L=\underset{n\to \mathrm{\infty }}{lim}\left|\frac{a_{n+1}}{a_n}\right|=\underset{n\to \mathrm{\infty }}{lim}\frac{(n+1{)}^2/2^{n+1}}{n^2/2^n}=\underset{n\to \mathrm{\infty }}{lim}\frac{(n+1{)}^2}{n^2}\cdot \frac{1}{2}=\frac{1}{2}<1$.

Since $L<1$, the series converges.

Step 3: Final Answer:
The series converges.

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Example 33: Divergence of a Series

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n+\sin (n)}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n+\sin (n)}$.

Step 2: Solution:
Compare with the harmonic series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$.
Since $n+\sin (n)\sim n$, the comparison shows that the series diverges.

Step 3: Final Answer:
The series diverges.

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Example 34: Limit Comparison Test with Convergence

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{3}{2n^2+1}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{3}{2n^2+1}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$.
Calculate the limit:
$L=\underset{n\to \mathrm{\infty }}{lim}\frac{\frac{3}{2n^2+1}}{\frac{1}{n^2}}=\underset{n\to \mathrm{\infty }}{lim}\frac{3n^2}{2n^2+1}=\frac{3}{2}$.

Since $L$ is finite and positive, both series converge together.
Thus, $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$ converges.

Step 3: Final Answer:
The series converges.

---

Example 35: Convergence of a Series with Factorials

Problem:
Determine the convergence of the series $\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}$.

Step 2: Solution:
Use the ratio test:
$L=\underset{n\to \mathrm{\infty }}{lim}\left|\frac{a_{n+1}}{a_n}\right|=\underset{n\to \mathrm{\infty }}{lim}\frac{x^{n+1}/(n+1)!}{x^n/n!}=\underset{n\to \mathrm{\infty }}{lim}\frac{|x|}{n+1}=0$.

Since $L<1$ for all $x$, the series converges.

Step 3: Final Answer:
The series converges for all $x$.

---

Example 36: Use of the Root Test

Problem:
Use the root test to determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{(3^n)}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{(3^n)}$.

Step 2: Solution:
Calculate the limit:
$L=\underset{n\to \mathrm{\infty }}{lim}\sqrt[n]{\frac{1}{3^n}}=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{3}=\frac{1}{3}<1$.

Since $L<1$, the series converges by the root test.

Step 3: Final Answer:
The series converges.

---

Example 37: Convergence of a Series with Square Root

Problem:
Determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{\sqrt{n}}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{\sqrt{n}}$ with $p=0.5$.

Step 2: Solution:
Since $p=0.5\le 1$, the series diverges.

Step 3: Final Answer:
The series diverges.

---

Example 38: Divergence Test for Series

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n}n$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n}n$.

Step 2: Solution:
Check the limit:
$\underset{n\to \mathrm{\infty }}{lim}n=\mathrm{\infty }$.

Since the limit does not approach zero, the series diverges.

Step 3: Final Answer:
The series diverges.

---

Example 39: Convergence of a Series with Cubes

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3}$ with $p=3$.

Step 2: Solution:
Since $p=3>1$, the series converges.

Step 3: Final Answer:
The series converges.

---

Example 40: Alternating Harmonic Series

Problem:
Determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n}$.

Step 2: Solution:
Check the terms:
Since $\frac{1}{n}$ is decreasing and $\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n}=0$, the series converges by the Alternating Series Test.

Step 3: Final Answer:
The series converges.

---

Example 41: Use of Comparison Test

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{2}{n^2+1}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{2}{n^2+1}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{2}{n^2}$.
Since $\frac{2}{n^2+1}<\frac{2}{n^2}$, and the comparison series converges, the original series converges.

Step 3: Final Answer:
The series converges.

---

Example 42: Divergence of a Series with Square Roots

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\sqrt{n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\sqrt{n}$.

Step 2: Solution:
The terms increase without bound:
$S_N=1+\sqrt{2}+\sqrt{3}+\dots +\sqrt{N}$ diverges.

Step 3: Final Answer:
The series diverges.

---

Example 43: Convergence of Series with Logarithm

Problem:
Determine the convergence of the series $\sum _{n=2}^{\mathrm{\infty }}\frac{\ln (n)}{n^2}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=2}^{\mathrm{\infty }}\frac{\ln (n)}{n^2}$.

Step 2: Solution:
Use the comparison test with $\sum _{n=2}^{\mathrm{\infty }}\frac{1}{n^{1.5}}$.
Since $\ln (n)$ grows slower than any polynomial, the series converges.

Step 3: Final Answer:
The series converges.

---

Example 44: Ratio Test for Series with Factorials

Problem:
Use the ratio test to determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{n!}{n^n}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{n!}{n^n}$.

Step 2: Solution:
Compute the ratio:
$L=\underset{n\to \mathrm{\infty }}{lim}\left|\frac{a_{n+1}}{a_n}\right|=\underset{n\to \mathrm{\infty }}{lim}\frac{(n+1)!}{(n+1{)}^{n+1}}\cdot \frac{n^n}{n!}$.

This simplifies to:
$=\underset{n\to \mathrm{\infty }}{lim}\frac{(n+1)n^n}{(n+1{)}^{n+1}}=0$.

Since $L<1$, the series converges.

Step 3: Final Answer:
The series converges.

---

Example 45: Divergence of an Oscillating Series

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n}n$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n}n$.

Step 2: Solution:
Check the limit:
$\underset{n\to \mathrm{\infty }}{lim}n=\mathrm{\infty }$.

Since the limit does not approach zero, the series diverges.

Step 3: Final Answer:
The series diverges.

---

Example 46: Absolute Convergence of Alternating Series

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n^3}$ converges absolutely.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n^3}$.

Step 2: Solution:
Check absolute convergence by evaluating $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3}$.
Since $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3}$ converges, the original series converges absolutely.

Step 3: Final Answer:
The series converges absolutely.

---

Example 47: Conditional Convergence

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n}$ converges conditionally.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n}$.

Step 2: Solution:
Check for conditional convergence:
The series converges by the Alternating Series Test.
However, the absolute series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$ diverges.

Step 3: Final Answer:
The series converges conditionally.

---

Example 48: Comparison Test for Series

Problem:
Use the comparison test to determine if the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{3n^2+1}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{3n^2+1}$.

Step 2: Solution:
Compare with the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$.
Since $\frac{1}{3n^2+1}<\frac{1}{n^2}$ and $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$ converges, the original series converges.

Step 3: Final Answer:
The series converges.

---

Example 49: Divergence of Series with Polynomial

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^4+n^2}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^4+n^2}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^4}$.
Since $n^4+n^2\sim n^4$ as $n\to \mathrm{\infty }$, the series converges.

Step 3: Final Answer:
The series converges.

---

Example 50: Limit Comparison with Divergence

Problem:
Use the limit comparison test to determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{n^2}{n^3+1}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{n^2}{n^3+1}$.

Step 2: Solution:
Calculate the limit:
$L=\underset{n\to \mathrm{\infty }}{lim}\frac{\frac{n^2}{n^3+1}}{\frac{1}{n}}=\underset{n\to \mathrm{\infty }}{lim}\frac{n^3}{n^3+1}=1$.

Since $L$ is finite and positive, the series diverges with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$.

Step 3: Final Answer:
The series diverges.

---

Example 51: Convergence of Series with Cubes

Problem:
Determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3+n}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3+n}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3}$.
Since $n^3+n\sim n^3$ as $n\to \mathrm{\infty }$, the series converges.

Step 3: Final Answer:
The series converges.

---

Example 52: Divergence of Series with Oscillating Terms

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n{n}^2$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n{n}^2$.

Step 2: Solution:
Check the limit:
$\underset{n\to \mathrm{\infty }}{lim}{n}^2=\mathrm{\infty }$.

Since the limit does not approach zero, the series diverges.

Step 3: Final Answer:
The series diverges.

---

Example 53: Convergence of Series with Exponential Functions

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n!}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n!}$.

Step 2: Solution:
Use the ratio test:
$L=\underset{n\to \mathrm{\infty }}{lim}\left|\frac{a_{n+1}}{a_n}\right|=\underset{n\to \mathrm{\infty }}{lim}\frac{1/(n+1)!}{1/n!}=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n+1}=0$.

Since $L<1$, the series converges.

Step 3: Final Answer:
The series converges.

---

Example 54: Series of Roots

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{\sqrt{n}}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{\sqrt{n}}$ with $p=0.5$.

Step 2: Solution:
Since $p=0.5\le 1$, the series diverges.

Step 3: Final Answer:
The series diverges.

---

Example 55: Series with Logarithmic Divergence

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{\ln (n)}{n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{\ln (n)}{n}$.

Step 2: Solution:
Use the integral test:
Consider the integral ${\int }_1^{\mathrm{\infty }}\frac{\ln (x)}{x},dx$.

Evaluating gives:
$=\frac{(\ln (x){)}^2}{2}{|}_1^{\mathrm{\infty }}=\mathrm{\infty }$.

Thus, the series diverges.

Step 3: Final Answer:
The series diverges.

---

Example 56: Convergence of Series with Factorials

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{n^n}{(n!{)}^2}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{n^n}{(n!{)}^2}$.

Step 2: Solution:
Use the ratio test:
$L=\underset{n\to \mathrm{\infty }}{lim}\left|\frac{a_{n+1}}{a_n}\right|$.

Evaluate:
$=\underset{n\to \mathrm{\infty }}{lim}\frac{(n+1{)}^{n+1}}{((n+1)!{)}^2}\cdot \frac{(n!{)}^2}{n^n}=\underset{n\to \mathrm{\infty }}{lim}\frac{(n+1{)}^{n+1}}{(n+1{)}^2{n}^n}=0$.

Since $L<1$, the series converges.

Step 3: Final Answer:
The series converges.

---

Example 57: Convergence of Series with Factorials

Problem:
Determine whether the series $\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^{n}n}{n!}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^{n}n}{n!}$.

Step 2: Solution:
Use the ratio test:
$L=\underset{n\to \mathrm{\infty }}{lim}|\frac{(-1{)}^{n+1}(n+1)}{(n+1)!}\cdot \frac{n!}{(-1{)}^{n}n}|$.

Simplifying gives:
$L=\underset{n\to \mathrm{\infty }}{lim}\frac{n+1}{(n+1)}=0$.

Since $L<1$, the series converges.

Step 3: Final Answer:
The series converges.

---

Example 58: Use of the Integral Test

Problem:
Determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n\ln (n)}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n\ln (n)}$.

Step 2: Solution:
Use the integral test:
Consider the integral ${\int }_2^{\mathrm{\infty }}\frac{1}{x\ln (x)},dx$.

This diverges by substitution $u=\ln (x)$.

Step 3: Final Answer:
The series diverges.

---

Example 59: Limit Comparison with Divergence

Problem:
Determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{n}{n^2+1}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{n}{n^2+1}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$.
Calculate the limit:
$L=\underset{n\to \mathrm{\infty }}{lim}\frac{\frac{n}{n^2+1}}{\frac{1}{n}}=\underset{n\to \mathrm{\infty }}{lim}\frac{n^2}{n^2+1}=1$.

Since $L$ is finite and positive, the original series diverges.

Step 3: Final Answer:
The series diverges.

---

Example 60: Series with Polynomial Growth

Problem:
Determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{n^2}{n^3+1}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{n^2}{n^3+1}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$.
Since $\frac{n^2}{n^3+1}\sim \frac{1}{n}$ as $n\to \mathrm{\infty }$, the series diverges.

Step 3: Final Answer:
The series diverges.

---

Example 61: Series with Roots

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{\sqrt{n^2+1}}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{\sqrt{n^2+1}}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$.
Since $\frac{1}{\sqrt{n^2+1}}\sim \frac{1}{n}$ as $n\to \mathrm{\infty }$, the series diverges.

Step 3: Final Answer:
The series diverges.

---

Example 62: Convergence of Logarithmic Series

Problem:
Determine the convergence of the series $\sum _{n=2}^{\mathrm{\infty }}\frac{1}{n\ln (n)}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=2}^{\mathrm{\infty }}\frac{1}{n\ln (n)}$.

Step 2: Solution:
Use the integral test:
Consider the integral ${\int }_2^{\mathrm{\infty }}\frac{1}{x\ln (x)},dx$.

This diverges, hence the series diverges.

Step 3: Final Answer:
The series diverges.

---

Example 63: Ratio Test for Series with Exponential Functions

Problem:
Determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{n^n}{n!}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{n^n}{n!}$.

Step 2: Solution:
Compute the ratio:
$L=\underset{n\to \mathrm{\infty }}{lim}\left|\frac{a_{n+1}}{a_n}\right|=\underset{n\to \mathrm{\infty }}{lim}\frac{(n+1{)}^{n+1}}{(n+1)!}\cdot \frac{n!}{n^n}$.

This simplifies to:
$L=\underset{n\to \mathrm{\infty }}{lim}\frac{(n+1)n^n}{(n+1{)}^{n+1}}=0$.

Since $L<1$, the series converges.

Step 3: Final Answer:
The series converges.

---

Example 64: Divergence of an Oscillating Series

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\sqrt{n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\sqrt{n}$.

Step 2: Solution:
Check the limit:
$\underset{n\to \mathrm{\infty }}{lim}\sqrt{n}=\mathrm{\infty }$.

Since the limit does not approach zero, the series diverges.

Step 3: Final Answer:
The series diverges.

---

Example 65: Convergence of Series with Factorials

Problem:
Determine the convergence of the series $\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(n+1)!}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{(n+1)!}$.

Step 2: Solution:
Check absolute convergence:
Evaluate $\sum _{n=0}^{\mathrm{\infty }}\frac{1}{(n+1)!}$, which converges.

Thus, the original series converges absolutely.

Step 3: Final Answer:
The series converges.

---

Example 66: Comparison of Series with Polynomial Growth

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2+n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2+n}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$.
Since $n^2+n\sim n^2$, the series converges.

Step 3: Final Answer:
The series converges.

---

Example 67: Divergence of Series with Factorials

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{n!}{n^n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{n!}{n^n}$.

Step 2: Solution:
Use the ratio test:
$L=\underset{n\to \mathrm{\infty }}{lim}|\frac{(n+1)!}{(n+1{)}^{n+1}}\cdot \frac{n^n}{n!}|$.

This simplifies to:
$=\underset{n\to \mathrm{\infty }}{lim}\frac{n^n(n+1)}{(n+1)(n^n)}=0$.

Since $L<1$, the series converges.

Step 3: Final Answer:
The series converges.

---

Example 68: Use of the Integral Test for Series

Problem:
Determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n\ln (n)}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n\ln (n)}$.

Step 2: Solution:
Use the integral test:
Consider the integral ${\int }_2^{\mathrm{\infty }}\frac{1}{x\ln (x)},dx$.

This diverges, hence the series diverges.

Step 3: Final Answer:
The series diverges.

---

Example 69: Convergence of a Series with Exponential Growth

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{2^n}{n!}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{2^n}{n!}$.

Step 2: Solution:
Use the ratio test:
$L=\underset{n\to \mathrm{\infty }}{lim}\left|\frac{a_{n+1}}{a_n}\right|=\underset{n\to \mathrm{\infty }}{lim}\frac{2^{n+1}}{(n+1)!}\cdot \frac{n!}{2^n}$.

This simplifies to:
$L=\underset{n\to \mathrm{\infty }}{lim}\frac{2}{n+1}=0$.

Since $L<1$, the series converges.

Step 3: Final Answer:
The series converges.

---

Example 70: Convergence of Series with Factorials

Problem:
Determine the convergence of the series $\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=0}^{\mathrm{\infty }}\frac{x^n}{n!}$.

Step 2: Solution:
Use the ratio test:
$L=\underset{n\to \mathrm{\infty }}{lim}\left|\frac{a_{n+1}}{a_n}\right|=\underset{n\to \mathrm{\infty }}{lim}\frac{x^{n+1}}{(n+1)!}\cdot \frac{n!}{x^n}=\underset{n\to \mathrm{\infty }}{lim}\frac{|x|}{n+1}=0$.

Since $L<1$, the series converges for all $x$.

Step 3: Final Answer:
The series converges for all $x$.

---

Example 71: Alternating Series with Logarithmic Terms

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n\ln (n)}{n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n\ln (n)}{n}$.

Step 2: Solution:
Check if the terms $\frac{\ln (n)}{n}$ are decreasing and if $\underset{n\to \mathrm{\infty }}{lim}\frac{\ln (n)}{n}=0$.
Both conditions are satisfied, so the series converges by the Alternating Series Test.

Step 3: Final Answer:
The series converges.

---

Example 72: Divergence of Series with Higher Power

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3+n^2}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3+n^2}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3}$.
Since $n^3+n^2\sim n^3$, the series converges.

Step 3: Final Answer:
The series converges.

---

Example 73: Limit Comparison Test with Harmonic Series

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n\ln (n)}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n\ln (n)}$.

Step 2: Solution:
Use the limit comparison test with the harmonic series:
$L=\underset{n\to \mathrm{\infty }}{lim}\frac{\frac{1}{n\ln (n)}}{\frac{1}{n}}=\underset{n\to \mathrm{\infty }}{lim}\frac{1}{\ln (n)}=0$.

Since $L=0$ and the harmonic series diverges, the series diverges.

Step 3: Final Answer:
The series diverges.

---

Example 74: Convergence of Series with Trigonometric Functions

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{\sin (n)}{n^2}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{\sin (n)}{n^2}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$.
Since $\sin (n)$ is bounded, the series converges by comparison.

Step 3: Final Answer:
The series converges.

---

Example 75: Divergence of Series with Logarithmic Terms

Problem:
Determine whether the series $\sum _{n=2}^{\mathrm{\infty }}\frac{1}{n\ln (n^2)}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=2}^{\mathrm{\infty }}\frac{1}{n\ln (n^2)}$.

Step 2: Solution:
Use the integral test:
Consider the integral ${\int }_2^{\mathrm{\infty }}\frac{1}{x\ln (x^2)},dx$.

This diverges, hence the series diverges.

Step 3: Final Answer:
The series diverges.

---

Example 76: Comparison of Series with Roots

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{\sqrt{n^3}}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{\sqrt{n^3}}$.

Step 2: Solution:
Since this is a $p$-series with $p=1.5>1$, the series converges.

Step 3: Final Answer:
The series converges.

---

Example 77: Alternating Series with Factorials

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n!}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n!}$.

Step 2: Solution:
Check the terms $a_n=\frac{1}{n!}$.
Since $a_n$ is decreasing and $\underset{n\to \mathrm{\infty }}{lim}{a}_n=0$, the series converges by the Alternating Series Test.

Step 3: Final Answer:
The series converges.

---

Example 78: Divergence of Series with Factorials

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}n!$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}n!$.

Step 2: Solution:
The terms increase without bound, hence the series diverges.

Step 3: Final Answer:
The series diverges.

---

Example 79: Limit Comparison with Convergence

Problem:
Use the limit comparison test to determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3+n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3+n}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3}$.
Calculate the limit:
$L=\underset{n\to \mathrm{\infty }}{lim}\frac{\frac{1}{n^3+n}}{\frac{1}{n^3}}=\underset{n\to \mathrm{\infty }}{lim}\frac{n^3}{n^3+n}=1$.

Since $L$ is finite and positive, the original series converges.

Step 3: Final Answer:
The series converges.

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Example 80: Series with Logarithmic Divergence

Problem:
Determine whether the series $\sum _{n=2}^{\mathrm{\infty }}\frac{\ln (n)}{n^2}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=2}^{\mathrm{\infty }}\frac{\ln (n)}{n^2}$.

Step 2: Solution:
Use the comparison test with $\sum _{n=2}^{\mathrm{\infty }}\frac{1}{n^{1.5}}$.
Since $\ln (n)$ grows slower than any polynomial, the series converges.

Step 3: Final Answer:
The series converges.

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Example 81: Convergence of Series with Trigonometric Functions

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{\sin (n)}{n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{\sin (n)}{n}$.

Step 2: Solution:
Since $\sin (n)$ is bounded, compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$.
The original series converges.

Step 3: Final Answer:
The series converges.

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Example 82: Divergence of Series with Factorials

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{n!}{2^n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{n!}{2^n}$.

Step 2: Solution:
Use the ratio test:
$L=\underset{n\to \mathrm{\infty }}{lim}\left|\frac{(n+1)!/2^{n+1}}{n!/2^n}\right|=\underset{n\to \mathrm{\infty }}{lim}\frac{(n+1)}{2}=\mathrm{\infty }$.

Since $L>1$, the series diverges.

Step 3: Final Answer:
The series diverges.

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Example 83: Series with Factorials

Problem:
Determine whether the series $\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{n}^2}{n!}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{n}^2}{n!}$.

Step 2: Solution:
Use the ratio test:
$L=\underset{n\to \mathrm{\infty }}{lim}\left|\frac{(-1{)}^{n+1}(n+1{)}^2/(n+1)!}{(-1{)}^n{n}^2/n!}\right|$.

This simplifies to:
$L=\underset{n\to \mathrm{\infty }}{lim}\frac{(n+1{)}^2}{(n+1)n^2}=0$.

Since $L<1$, the series converges.

Step 3: Final Answer:
The series converges.

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Example 84: Divergence of Series with Powers

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^4+n^3}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^4+n^3}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^4}$.
Since $n^4+n^3\sim n^4$, the series converges.

Step 3: Final Answer:
The series converges.

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Example 85: Limit Comparison with Divergence

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2+n+1}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2+n+1}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$.
Calculate the limit:
$L=\underset{n\to \mathrm{\infty }}{lim}\frac{\frac{1}{n^2+n+1}}{\frac{1}{n^2}}=\underset{n\to \mathrm{\infty }}{lim}\frac{n^2}{n^2+n+1}=1$.

Since $L$ is finite and positive, the original series converges.

Step 3: Final Answer:
The series converges.

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Example 86: Convergence of Series with Trigonometric Functions

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{\cos (n)}{n^2}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{\cos (n)}{n^2}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$.
Since $\cos (n)$ is bounded, the series converges by comparison.

Step 3: Final Answer:
The series converges.

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Example 87: Alternating Series Test

Problem:
Determine whether the alternating series $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n^3}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n}{n^3}$.

Step 2: Solution:
Check the terms:
Since $\frac{1}{n^3}$ is decreasing and $\underset{n\to \mathrm{\infty }}{lim}\frac{1}{n^3}=0$, the series converges by the Alternating Series Test.

Step 3: Final Answer:
The series converges.

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Example 88: Divergence of Series with Increasing Terms

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n}n$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^{n}n$.

Step 2: Solution:
Check the limit:
$\underset{n\to \mathrm{\infty }}{lim}n=\mathrm{\infty }$.

Since the limit does not approach zero, the series diverges.

Step 3: Final Answer:
The series diverges.

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Example 89: Comparison with Harmonic Series

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n\ln (n)}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n\ln (n)}$.

Step 2: Solution:
Use the integral test:
Consider the integral ${\int }_2^{\mathrm{\infty }}\frac{1}{x\ln (x)},dx$.

This diverges, hence the series diverges.

Step 3: Final Answer:
The series diverges.

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Example 90: Series with Polynomial Growth

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{n^2}{n^3+1}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{n^2}{n^3+1}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}$.
Since $\frac{n^2}{n^3+1}\sim \frac{1}{n}$ as $n\to \mathrm{\infty }$, the series diverges.

Step 3: Final Answer:
The series diverges.

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Example 91: Limit Comparison Test

Problem:
Use the limit comparison test to determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3+n^2}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3+n^2}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3}$.
Calculate the limit:
$L=\underset{n\to \mathrm{\infty }}{lim}\frac{\frac{1}{n^3+n^2}}{\frac{1}{n^3}}=\underset{n\to \mathrm{\infty }}{lim}\frac{n^3}{n^3+n^2}=1$.

Since $L$ is finite and positive, the original series converges.

Step 3: Final Answer:
The series converges.

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Example 92: Divergence of Series with Oscillating Terms

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n{n}^2}{n!}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n{n}^2}{n!}$.

Step 2: Solution:
Check the terms:
The series converges by the ratio test:
$L=\underset{n\to \mathrm{\infty }}{lim}\frac{(n+1{)}^2}{(n+1)!}\cdot \frac{n!}{n^2}=0$.

Step 3: Final Answer:
The series converges.

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Example 93: Convergence of Series with Exponential Growth

Problem:
Determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{n^n}{n!}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{n^n}{n!}$.

Step 2: Solution:
Use the ratio test:
$L=\underset{n\to \mathrm{\infty }}{lim}|\frac{(n+1{)}^{n+1}}{(n+1)!}\cdot \frac{n!}{n^n}|$.

This simplifies to:
$L=\underset{n\to \mathrm{\infty }}{lim}\frac{(n+1{)}^n}{(n+1{)}^{n+1}}=\mathrm{\infty }$.

Since $L>1$, the series diverges.

Step 3: Final Answer:
The series diverges.

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Example 94: Divergence of Series with Polynomial Growth

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2+n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2+n}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^2}$.
Since $n^2+n\sim n^2$, the series converges.

Step 3: Final Answer:
The series converges.

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Example 95: Limit Comparison Test

Problem:
Use the limit comparison test to determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^4+1}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^4+1}$.

Step 2: Solution:
Compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^4}$.
Calculate the limit:
$L=\underset{n\to \mathrm{\infty }}{lim}\frac{\frac{1}{n^4+1}}{\frac{1}{n^4}}=1$.

Since $L$ is finite and positive, the original series converges.

Step 3: Final Answer:
The series converges.

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Example 96: Divergence of Series with Oscillating Terms

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\frac{1}{n^2+n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\frac{1}{n^2+n}$.

Step 2: Solution:
The terms converge to zero, hence the series converges by the Alternating Series Test.

Step 3: Final Answer:
The series converges.

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Example 97: Use of Integral Test

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^p}$ converges for $p=1.5$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^{1.5}}$.

Step 2: Solution:
Since $p=1.5>1$, the series converges.

Step 3: Final Answer:
The series converges.

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Example 98: Series with Trigonometric Functions

Problem:
Determine the convergence of the series $\sum _{n=1}^{\mathrm{\infty }}\frac{\sin (n)}{n^3}$.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{\sin (n)}{n^3}$.

Step 2: Solution:
Since $|\sin (n)|\le 1$, compare with $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^3}$.
The series converges.

Step 3: Final Answer:
The series converges.

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Example 99: Divergence of Series with Factorials

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{n!}{2^n}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{n!}{2^n}$.

Step 2: Solution:
Use the ratio test:
$L=\underset{n\to \mathrm{\infty }}{lim}\left|\frac{(n+1)!/2^{n+1}}{n!/2^n}\right|=\underset{n\to \mathrm{\infty }}{lim}\frac{(n+1)}{2}=\mathrm{\infty }$.

Since $L>1$, the series diverges.

Step 3: Final Answer:
The series diverges.

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Example 100: Convergence of Series with Logarithmic Functions

Problem:
Determine whether the series $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n\ln (n^2)}$ converges or diverges.

Answer:

Step 1: Given Data:
The series is $\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n\ln (n^2)}$.

Step 2: Solution:
Use the integral test:
Consider the integral ${\int }_2^{\mathrm{\infty }}\frac{1}{x\ln (x^2)},dx$.

This diverges, hence the series diverges.

Step 3: Final Answer:
The series diverges.