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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{\ln(n)} } }\)
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.
video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(1)^n}{\sqrt{n+1}} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=3}^{\infty}{ \frac{1}{n^2 \ln n} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n}e^{\sqrt{n}}} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^n}{3^{n^2}} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ n \sin(1/n) } }\)
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.
video by blackpenredpen 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n+3^n} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(1)^n}{3n+1} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{3^{\ln n}} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3n^2+n}{\sqrt{n^5+2n+1}} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{1+1/n}} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n^{1+1/n^2}} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ 1 } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ (11/n)^n } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ (11/n)^n } }\) converges or diverges.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ (11/n)^{n^2} } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ (11/n)^{n^2} } }\) converges or diverges.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sin^4 n} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n!}{n^n} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 counterexample.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 counterexample.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 counterexample.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 counterexample.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 counterexample.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 counterexample.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 counterexample.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 counterexample.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 counterexample.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ (1  \cos(1/n)) } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt[3]{2^n+1}} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=2}^{\infty}{ \frac{2}{\sqrt{n}\ln n} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n1}{\sqrt{n^3+2n+5}} } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n1}{\sqrt{n^3+2n+5}} } }\) converges or diverges.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^2 2^{n+2}}{4^n} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

If possible, evaluate \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }\) where \( a_1 = 9 \) and \( a_n = (6n)a_{n1} \) for \( n \geq 2 \).
Problem Statement 

If possible, evaluate \(\displaystyle{ \sum_{n=1}^{\infty}{ a_n } }\) where \( a_1 = 9 \) and \( a_n = (6n)a_{n1} \) for \( n \geq 2 \).
Solution 

Finishing the problem, we have \( 9 + 36 + 108 + 216 + 216 = 585 \)
video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ e^{n}\sin(n) } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^{10}4^n}{n!} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n!}} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^3}{2n^5+3n4} } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^3}{2n^5+3n4} } }\) converges or diverges.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{12n}{3+4n} \right]^n } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \left[ \frac{12n}{3+4n} \right]^n } }\) converges or diverges.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{e^n}{2^{2n1}} } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{e^n}{2^{2n1}} } }\) converges or diverges.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^32n1}{2n^5+3n4} } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{n^32n1}{2n^5+3n4} } }\) converges or diverges.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n+\sqrt{n}} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{n\sqrt{n}} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3^n n^2}{n!} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

For what values of \(x\) will the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x2)^n}{n^n} } }\) converge?
Problem Statement 

For what values of \(x\) will the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x2)^n}{n^n} } }\) converge?
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

For what values of \(x\) will the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x1)^n}{n \cdot 3^n} } }\) converge?
Problem Statement 

For what values of \(x\) will the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{(x1)^n}{n \cdot 3^n} } }\) converge?
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\sqrt{n}  \sqrt{n+1}} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{e^{\sqrt{n}}} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{n\sqrt{n^51}} } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=2}^{\infty}{ \frac{1}{n\sqrt{n^51}} } }\) converges or diverges.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{e^n+1} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\ln(e^n1)} } }\)
Problem Statement 

Determine whether the series \(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{1}{\ln(e^n1)} } }\) converges or diverges.
Solution 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

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 

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \sum_{n=1}^{\infty}{ \frac{3^n n!}{n^n} } }\)
Problem Statement 

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

video by blackpenredpen 

close solution

Log in to rate this practice problem and to see it's current rating. 

You CAN Ace Calculus
all infinite series topics 
external links you may find helpful 

To bookmark this page and practice problems, log in to your account or set up a free account.
Single Variable Calculus 

MultiVariable Calculus 

Differential Equations 

Precalculus 

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
 
Help Keep 17Calculus Free 

The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free. 
You Can Have an Amazing Memory: Learn LifeChanging Techniques and Tips from the Memory Maestro 

Practice Instructions
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.