# How to choose a convergence test for a given infinite series?

I have a general question when it comes to deciding if an infinite series is convergent or divergent. The tests im familiar with are ; Ratio test, Direct comparison test, Limit comparison test, Root test and the Integral test.

My question is if there is any way to tell what test is appropriate to start with just by looking at the series. At the moment I usually follow my own pattern and systematically try different tests. What I do is:

1. Divergence test to see if the series is divergent. If its unsuccessful
2. Ratio test, if unsuccessful
3. Limit comparison test
4. etc

Which has been working fine, my concern is that it may be very time consuming if the first tests are unsuccessful, and might not be very efficient during exams.

So basically: Is there a way to determine what tests are appropriate by just looking at the series? ( I have found nothing like this in my textbook, all the examples simply jump straight into the "correct" test).

## 1 Answer

We have learned that

• ratio test is the best for series that converges to zero "fastly" - i.e. when you see that the series has for example factorial, exponential function or something like that in the denominator, it's good to try ratio test ($\sum{\frac{n^5}{5^n}}$,...)
• limit comparison test is good for series that converges to 0 slowly that means n-th roots, polynomial function etc ($\sum{\frac{n^2}{n^3+1}...}$) then you just have to find a right series to compare it to
• root test is good for series containing exponential functions ($\sum ({\frac{3n+42}{4n+2})^n}$...)

When solving convergence of a series it´s hard to say if one test is better than the others. Different tests can be used to solve the same series. But I hope that those are some useful tips