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?

  • 3
    $\begingroup$ In my experience, this is largely a skill one develops over time. $\endgroup$ – Alex Becker Jul 18 '12 at 16:39
  • 1
    $\begingroup$ 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.) $\endgroup$ – Qiaochu Yuan Jul 18 '12 at 16:52

The p-series and geometric series tests are for specific types of sequences, and it is clear when you can apply them.

Use the integral test for positive, decreasing functions or negative, increasing functions only (do not forget this condition).

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.

The best thing to do is to simply practice exercises until you have mastered their usage.

  • 1
    $\begingroup$ 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 $\sum \frac{1}{k \log^2 k}$. $\endgroup$ – Qiaochu Yuan Jul 18 '12 at 19:14
  • $\begingroup$ If the function is positive and increasing, the series diverges by the divergence test, so there is no need for integral test. $\endgroup$ – user147263 Jan 4 '15 at 15:35

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