## 17Calculus Infinite Series - 100 Practice Problems

##### 17Calculus

This page consists of 100 (actually 101) infinite series practice problems based on a video from one of our favorite instructors. We have laid out each practice problem and included the video clip containing each solution.

Here is the list of practice problems. We recommend that you download this pdf before starting.

Make sure you support the guy that did this video. He put a LOT of work into, not just doing the video, but also preparing the problems and making sure his solutions were correct. He did a GREAT job. So go to YouTube and like this video and follow him. He is one of our favorite instructors. (By the way, we are not receiving any compensation from him. We just think his videos will help you.)

We have another page with 100 calculus 2 practice problems that also contains some infinite series problems by this same guy.

Practice

Unless otherwise instructed, determine the convergence or divergence of these series. If possible, determine the value to which the series converges and whether the series converges conditionally or absolutely. Make sure to specify the test(s) and theorem(s) you used as part of your final answer.

Questions 1 - 10

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{\ln(n)} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{\ln(n)} } }$$ converges or diverges.

Solution

Practice problem 157 on the Direct Comparison Test page shows the details on proving that $$1/\ln(n) \geq 1/n$$. We do not consider his logic in 'The List' to be adequate for showing divergence, although we do recommend that you confirm your answer this way.

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$$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{\ln(n^n)} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{\ln(n^n)} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1619}^{\infty}{ \frac{1}{(\ln n)^{\ln n}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1619}^{\infty}{ \frac{1}{(\ln n)^{\ln n}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{\tan^{-1}n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{\tan^{-1}n} } }$$ converges or diverges.

Hint

In this context, $$\tan^{-1}n = \arctan(n)$$.

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{\tan^{-1}n} } }$$ converges or diverges.

Hint

In this context, $$\tan^{-1}n = \arctan(n)$$.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2^n}{3^n+n^3} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2^n}{3^n+n^3} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3^n}{2^n+n^2} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3^n}{2^n+n^2} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n \sin^2 n}{n^3 + 2} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n \sin^2 n}{n^3 + 2} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{\sqrt{n+1}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{\sqrt{n+1}} } }$$ converges or diverges.

Solution

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If possible, evaluate $$1/2 - 1/3 + 2/9 - 4/27 + \cdots$$.

Problem Statement

If possible, evaluate $$1/2 - 1/3 + 2/9 - 4/27 + \cdots$$.

Solution

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Questions 11 - 20

$$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{1}{\sqrt{n}}- \frac{1}{n} \right] } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{1}{\sqrt{n}}- \frac{1}{n} \right] } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=3}^{\infty}{ \frac{1}{n^2 \ln n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=3}^{\infty}{ \frac{1}{n^2 \ln n} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n}e^{\sqrt{n}}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n}e^{\sqrt{n}}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^n}{3^{n^2}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^n}{3^{n^2}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^n}{(n!)^2} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^n}{(n!)^2} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ n \sin(1/n) } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ n \sin(1/n) } }$$ converges or diverges.

Solution

The first video below is the video clip that solves this problem. During that solution, he mentions a video about the limit $$\displaystyle{ \lim_{ heta o 0}{ rac{\sin(\theta)}{\theta} } }$$. The second video below is that video with lots of explanation.

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n+3^n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n+3^n} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\sin(2n)}{n+3^n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\sin(2n)}{n+3^n} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{3n+1} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{3n+1} } }$$ converges or diverges.

Solution

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If possible, evaluate $$1/2 + 1/6 + 1/12 + 1/20 + \cdots$$.

Problem Statement

If possible, evaluate $$1/2 + 1/6 + 1/12 + 1/20 + \cdots$$.

Solution

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Questions 21 - 30

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n!}{e^{n^2}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n!}{e^{n^2}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2+1}{n^3+1} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2+1}{n^3+1} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \sin(1/n^2) } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \sin(1/n^2) } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \cos^2(1/n) } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \cos^2(1/n) } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\cos(\pi n)}{\ln n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\cos(\pi n)}{\ln n} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(2n+1)^n}{n^{2n}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(2n+1)^n}{n^{2n}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{2^{\ln n}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{2^{\ln n}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{3^{\ln n}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{3^{\ln n}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3n^2+n}{\sqrt{n^5+2n+1}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3n^2+n}{\sqrt{n^5+2n+1}} } }$$ converges or diverges.

Solution

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Evaluate $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n}{2^n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n}{2^n} } }$$ converges or diverges. If it converges, find the sum, if possible.

Solution

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Questions 31 - 40

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(n!)^2}{(2n)!} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(n!)^2}{(2n)!} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{1+1/n}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{1+1/n}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{1+1/n^2}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{1+1/n^2}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ 1 } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ 1 } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2}{2^n+3^n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2}{2^n+3^n} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ (1-1/n)^n } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ (1-1/n)^n } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ (1-1/n)^{n^2} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ (1-1/n)^{n^2} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sin^4 n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sin^4 n} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n!}{n^n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n!}{n^n} } }$$ converges or diverges.

Solution

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Evaluate $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^3+3n^2+2n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^3+3n^2+2n} } }$$ converges or diverges. If it converges, find the sum, if possible.

Solution

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Questions 41 - 50

If $$\displaystyle{ \sum_{n=1}^{\infty}{ (a_n)^2 } }$$ converges, then $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ must also converge. Is this true or false? Explain.

Problem Statement

If $$\displaystyle{ \sum_{n=1}^{\infty}{ (a_n)^2 } }$$ converges, then $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ must also converge. Is this true or false? If it is true, prove it. If it is false, provide a counter-example.

Solution

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If $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ converges, then $$\displaystyle{ \sum_{n=1}^{\infty}{ (a_n)^2 } }$$ must also converge. Is this true or false? Explain.

Problem Statement

If $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ converges, then $$\displaystyle{ \sum_{n=1}^{\infty}{ (a_n)^2 } }$$ must also converge. Is this true or false? If it is true, prove it. If it is false, provide a counter-example.

Solution

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If $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ converges, then $$\displaystyle{ \sum_{n=1}^{\infty}{ 1/a_n } }$$ must diverge. Is this true or false? Explain.

Problem Statement

If $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ converges, then $$\displaystyle{ \sum_{n=1}^{\infty}{ 1/a_n } }$$ must diverge. Is this true or false? If it is true, prove it. If it is false, provide a counter-example.

Solution

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If $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ diverges, then $$\displaystyle{ \sum_{n=1}^{\infty}{ 1/a_n } }$$ must converge. Is this true or false? Explain.

Problem Statement

If $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ diverges, then $$\displaystyle{ \sum_{n=1}^{\infty}{ 1/a_n } }$$ must converge. Is this true or false? If it is true, prove it. If it is false, provide a counter-example.

Solution

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If $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ converges, then $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n / n } }$$ must also converge. Is this true or false? Explain.

Problem Statement

If $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ converges, then $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n / n } }$$ must also converge. Is this true or false? If it is true, prove it. If it is false, provide a counter-example.

Solution

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If $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ and $$\displaystyle{ \sum_{n=1}^{\infty}{ b_n } }$$ both converge, then $$\displaystyle{ \sum_{n=1}^{\infty}{ (a_n + b_n) } }$$ must also converge. Is this true or false? Explain.

Problem Statement

If $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ and $$\displaystyle{ \sum_{n=1}^{\infty}{ b_n } }$$ both converge, then $$\displaystyle{ \sum_{n=1}^{\infty}{ (a_n + b_n) } }$$ must also converge. Is this true or false? If it is true, prove it. If it is false, provide a counter-example.

Solution

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If $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ and $$\displaystyle{ \sum_{n=1}^{\infty}{ b_n } }$$ both diverge and $$a_n \neq b_n$$, then $$\displaystyle{ \sum_{n=1}^{\infty}{ (a_n - b_n) } }$$ must also diverge. Is this true or false? Explain.

Problem Statement

If $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ and $$\displaystyle{ \sum_{n=1}^{\infty}{ b_n } }$$ both diverge and $$a_n \neq b_n$$, then $$\displaystyle{ \sum_{n=1}^{\infty}{ (a_n - b_n) } }$$ must also diverge. Is this true or false? If it is true, prove it. If it is false, provide a counter-example.

Solution

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If $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ and $$\displaystyle{ \sum_{n=1}^{\infty}{ b_n } }$$ both diverge, then $$\displaystyle{ \sum_{n=1}^{\infty}{ (a_n b_n) } }$$ must also diverge. Is this true or false? Explain.

Problem Statement

If $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ and $$\displaystyle{ \sum_{n=1}^{\infty}{ b_n } }$$ both diverge, then $$\displaystyle{ \sum_{n=1}^{\infty}{ (a_n b_n) } }$$ must also diverge. Is this true or false? If it is true, prove it. If it is false, provide a counter-example.

Solution

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If $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ and $$\displaystyle{ \sum_{n=1}^{\infty}{ b_n } }$$ both converge, then $$\displaystyle{ \sum_{n=1}^{\infty}{ (a_n b_n) } }$$ must also converge. Is this true or false? Explain.

Problem Statement

If $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ and $$\displaystyle{ \sum_{n=1}^{\infty}{ b_n } }$$ both converge, then $$\displaystyle{ \sum_{n=1}^{\infty}{ (a_n b_n) } }$$ must also converge. Is this true or false? If it is true, prove it. If it is false, provide a counter-example.

Solution

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Evaluate $$\displaystyle{ \sum_{n=1}^{\infty}{ 0 } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ 0 } }$$ converges or diverges. If it converges, find the sum, if possible.

Solution

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Questions 51 - 60

$$\displaystyle{ \sum_{n=1}^{\infty}{ n \sqrt{\sin(1/n^2)} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ n \sqrt{\sin(1/n^2)} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ (1 - \sin(1/n)) } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ (1 - \sin(1/n)) } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ (1 - \cos(1/n)) } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ (1 - \cos(1/n)) } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt[3]{2^n+1}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt[3]{2^n+1}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{2}{\sqrt{n}\ln n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{2}{\sqrt{n}\ln n} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n-1}{\sqrt{n^3+2n+5}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n-1}{\sqrt{n^3+2n+5}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2 2^{n+2}}{4^n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2 2^{n+2}}{4^n} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \sqrt[n]{2} - 1 } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \sqrt[n]{2} - 1 } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ (\sqrt[n]{2} - 1)^n } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ (\sqrt[n]{2} - 1)^n } }$$ converges or diverges.

Solution

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If possible, evaluate $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ where $$a_1 = 9$$ and $$a_n = (6-n)a_{n-1}$$ for $$n \geq 2$$.

Problem Statement

If possible, evaluate $$\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }$$ where $$a_1 = 9$$ and $$a_n = (6-n)a_{n-1}$$ for $$n \geq 2$$.

Solution

Finishing the problem, we have $$9 + 36 + 108 + 216 + 216 = 585$$

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Questions 61 - 70

$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(n!)^n}{n^{10n}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(n!)^n}{n^{10n}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(2n)!}{n^n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(2n)!}{n^n} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ e^{-n}\sin(n) } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ e^{-n}\sin(n) } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\tan(1/n)}{n^2} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{\tan(1/n)}{n^2} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^{10}4^n}{n!} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^{10}4^n}{n!} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2^n n!}{(n+2)!} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{2^n n!}{(n+2)!} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n!}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n!}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\ln(n!)} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\ln(n!)} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n(n+2)}{(2n+1)^2} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n(n+2)}{(2n+1)^2} } }$$ converges or diverges.

Solution

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If possible, evaluate $$\displaystyle{ \sum_{n=1}^{\infty}{ (e^{1/n} - e^{1/(n+2)}) } }$$.

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ (e^{1/n} - e^{1/(n+2)}) } }$$ converges or diverges. If it converges, find the sum, if possible.

Solution

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Questions 71 - 80

$$\displaystyle{ \sum_{n=3}^{\infty}{ \frac{\ln n}{n^2} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=3}^{\infty}{ \frac{\ln n}{n^2} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^3}{2n^5+3n-4} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^3}{2n^5+3n-4} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{1-2n}{3+4n} \right]^n } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{1-2n}{3+4n} \right]^n } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{e^n}{2^{2n-1}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{e^n}{2^{2n-1}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^3-2n-1}{2n^5+3n-4} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^3-2n-1}{2n^5+3n-4} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n+\sqrt{n}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n+\sqrt{n}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n\sqrt{n}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n\sqrt{n}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \sqrt{\cos(1/n)} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \sqrt{\cos(1/n)} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3^n n^2}{n!} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3^n n^2}{n!} } }$$ converges or diverges.

Solution

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If possible, evaluate $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n}{2^n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n}{2^n} } }$$ converges or diverges. If it converges, find the sum, if possible.

Solution

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Questions 81 - 90

For what values of $$x$$ will the series $$1^x + 2^x + 3^x + 4^x + \cdots + n^x + \cdots$$ converge?

Problem Statement

For what values of $$x$$ will the series $$1^x + 2^x + 3^x + 4^x + \cdots + n^x + \cdots$$ converge?

Solution

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For what values of $$x$$ will the series $$x^1 + x^2 + x^3 + x^4 + \cdots + x^n + \cdots$$ converge?

Problem Statement

For what values of $$x$$ will the series $$x^1 + x^2 + x^3 + x^4 + \cdots + x^n + \cdots$$ converge?

Solution

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For what values of $$x$$ will the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x-2)^n}{n^n} } }$$ converge?

Problem Statement

For what values of $$x$$ will the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x-2)^n}{n^n} } }$$ converge?

Solution

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For what values of $$x$$ will the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{x^n}{n} } }$$ converge?

Problem Statement

For what values of $$x$$ will the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{x^n}{n} } }$$ converge?

Solution

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For what values of $$x$$ will the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x-1)^n}{n \cdot 3^n} } }$$ converge?

Problem Statement

For what values of $$x$$ will the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x-1)^n}{n \cdot 3^n} } }$$ converge?

Solution

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For what values of $$x$$ will the series $$\displaystyle{ \sum_{n=1}^{\infty}{ n! x^n } }$$ converge?

Problem Statement

For what values of $$x$$ will the series $$\displaystyle{ \sum_{n=1}^{\infty}{ n! x^n } }$$ converge?

Solution

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For what values of $$k$$ will the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{x(\ln x)^k} } }$$ converge?

Problem Statement

For what values of $$k$$ will the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{x(\ln x)^k} } }$$ converge?

Solution

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For what values of $$x$$ will the series $$\displaystyle{ \sum_{n=0}^{\infty}{ \left[ \frac{1}{1-x} \right]^n } }$$ converge?

Problem Statement

For what values of $$x$$ will the series $$\displaystyle{ \sum_{n=0}^{\infty}{ \left[ \frac{1}{1-x} \right]^n } }$$ converge?

Solution

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For what values of $$x$$ will the series $$\displaystyle{ \sum_{n=0}^{\infty}{ \left[ \sum_{m=0}^{\infty}{ x^m } \right]^n } }$$ converge?

Problem Statement

For what values of $$x$$ will the series $$\displaystyle{ \sum_{n=0}^{\infty}{ \left[ \sum_{m=0}^{\infty}{ x^m } \right]^n } }$$ converge?

Solution

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If possible, evaluate $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{n!} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(-1)^n}{n!} } }$$ converges or diverges. If it converges, find the sum, if possible.

Solution

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Questions 91 - 101

$$\displaystyle{ \sum_{n=1}^{\infty}{ (\pi/2 - \tan^{-1} n) } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ (\pi/2 - \tan^{-1} n )} }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \sin^2(1/n) } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \sin^2(1/n) } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n} - \sqrt{n+1}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n} - \sqrt{n+1}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}} \right] } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}} \right] } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{e^{\sqrt{n}}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{e^{\sqrt{n}}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{n\sqrt{n^5-1}} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{n\sqrt{n^5-1}} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \ln[ n/(n+2) ] } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \ln[ n/(n+2) ] } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{e^n+1} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{e^n+1} } }$$ converges or diverges.

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\ln(e^n-1)} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\ln(e^n-1)} } }$$ converges or diverges.

Solution

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If possible, evaluate $$1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + \cdots$$

Problem Statement

If possible, evaluate $$1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + \cdots$$

Solution

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$$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3^n n!}{n^n} } }$$

Problem Statement

Determine whether the series $$\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3^n n!}{n^n} } }$$ converges or diverges.

Solution

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