# How can I determine which series comparison test to use?

In my textbook, there is a section of questions that's instructions reads "Test for convergence or divergence, using each one of the following tests once," and the test choices it gives me are

1. nth-Term Test
2. p-Series Test
3. Integral Test
4. Limit Comparison Test
5. Geometric Series Test
6. Telescoping Series Test
7. Direct Comparison Test

Now, I was able to get most of the problems in this section right on the first try, but when I compared my answers to those in the solutions manual, I found that often the way I chose was not the same as theirs. Sometimes, my solution was simpler, but most of the time, the method they chose was quicker. For one of the problems, I did the limit comparison test and I ended up with a $b_{n}$ that was one, so I was really just doing some convoluted mix between the Limit Comparison Test and the nth-Term Test.

How can I develop skills to help me to decide which method to choose? Is there some "trick" to deciding which method will be the simplest?

• In my experience, this is largely a skill one develops over time. – Alex Becker Jul 18 '12 at 16:39
• Practice. Experience. The same way you determine which tool you use to fix a car or build a chair. (The only difference is that these tools can be carried around in your head.) – Qiaochu Yuan Jul 18 '12 at 16:52

Telescoping series always look like $\sum f(x+1)-f(x)$, so like the other series, they are for a particular type of series but watch out for the series $\sum \frac{1}{n(n+1)}$ and similar series that can be made into a telescoping series using partial fractions.
• Note that to use the integral test you don't need to integrate the function itself, but (together with limit comparison and/or direct comparison) you only need to integrate functions which bound your function. For example it is hard to use the integral test on $\sum \frac{1}{k \log^2 k + 1}$ but it is easy to use it on the related sum $\sum \frac{1}{k \log^2 k}$. – Qiaochu Yuan Jul 18 '12 at 19:14