# How would I know when to use what test for convergence?

I'm specifically looking at the following convergence tests for series:

• Limit Comparison Theorem
• Direct Comparison Theorem
• Integral Test
• Ratio Test
• Root Test

I am currently struggling with determining when to use which convergence tests. Any help would be appreciated!

• I think you can try a few of them in A LOT of problems and, with enough practice, you will eventually develop an intuition on what tool is best for each situation. – Anna SdTC Nov 27 '17 at 3:11

Suppose you have a series $$\sum a_i$$ and you want to prove its convergence or divergence. Let's see when each test would be appropriate:
Limit comparison and direct comparison most effective if the terms in your sequence resemble a familiar series, such as a $$p$$-series. For example, suppose $$a_i = \frac{1}{i^3 + 2i}$$ which looks suspiciously looks like a convergent $$p$$-series (where $$p = 3$$). Indeed, $$\lim_{i \to \infty} \frac{i^3 + 2i}{i^3} = 1 \in (0, \infty)$$ so $$a_i$$ converges. Limit comparison is especially good for verifying series that intuitively seem like they should converge or diverge but it's hard to prove directly.
If $$a_i$$ looks like a function $$f(i)$$ whose integral you are comfortable computing, you should use the integral test. For example, suppose you didn't know the $$p$$-series test; then it would be easy to check that $$a_i = i^{-p}$$ converges if and only if $$p > 1$$simply by computing the integral $$\int_0^{\infty} x^{-p} ~dx$$ which isn't particularly hard. Be especially on the lookout for functions which look like they could be integrated by parts or $$u$$-substitution.
The ratio and root tests are more situational. The root test is mostly useful if $$a_i$$ is of the form $$a_i = b_i^i$$ for some sequence $$b_i$$ which you can easily compute the limit of, because then $$a_i^{1/i} = b_i$$ and one only has to compute $$\limsup b_i$$. Of course, if $$\limsup b_i = 1$$ then this test is inconclusive and this test is a red herring. Similarly, the ratio test is good for series whose terms can be easily cancelled by multiplication and division, but could be a red herring.