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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.

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This page contains a list of infinite series practice problems. These problems are not in any special order, similar to how you might find them on an exam. We work these problems multiple times with different tests to compare which might be best and why.

There are hundreds more practice problems on pages dedicated to specific infinite series techniques. Select a topic from the menu to go to a specific topic page.

Determining convergence or divergence of a series can often be done using several different tests. There is usually not just one way to work an infinite series problem. However, there is usually a best way. The practice problems listed here have extremely detailed solutions showing many possible ways to determine convergence or divergence. This allows you to compare the methods and begin to see patterns. The best way to learn to solve these problems efficiently is to start with the guidelines below and then tweak them as you get experience. Don't be afraid to make mistakes or go down rabbit trails to dead-ends. These are valuable learning experiences that will help you prepare for exams.

Practice Problems

There are hundreds of practice problems scattered around on the various infinite series pages. Since infinite series problems can usually be solved several ways, the problems below combine all the solutions so that you can compare techniques. To study a specific technique, select a topic from the menu to go to the page discussing that technique.

Instructions:
1. Determine the convergence or divergence of the following series.
2. If possible, determine the value the series converges to.
3. Determine if a convergent series converges absolutely or conditionally.

Caveat - This list of practice problems is in its infancy. After looking at one or two solutions, you can tell that these take significant time to develop and layout. We plan to add more of these in the future but for now, their value is to demonstrate a suggested way to determine which test to use.

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

\(\displaystyle{\sum_{n=0}^{\infty}{\frac{1}{3^n+n}}}\)

\(\displaystyle{\sum_{n=1}^{\infty}{\frac{n}{n^2-\cos^2(n)}}}\)

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

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

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

\(\displaystyle{\sum_{n=1}^{\infty}{\frac{n~\cos(n\pi)}{2n-1}}}\)

\(\displaystyle{ \sum_{k=1}^{\infty}{ \left[ \frac{1}{k+1} - \frac{1}{k+2} \right] } }\)

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