Which Test to Use: Ratio or Root? Understanding the Convergence of Series

[MissP.25_5]

Hello.
How do I determine whether to use ratio test or root test in determining whether a series is convergent or divergant? For example, in this problem, ratio is used for no.1 and root test for no.2. Why is that? I need explanation, please.

 

[Fightfish]

I would say that whichever test works is the test that you apply it to. It all boils down to the form of the terms in the series that you are interested in -- for certain forms, some tests may not be able to arrive at conclusive results or may be extremely inconvenient.

The first series that you listed down, $\frac{n!^{2}}{(2n)!}$, is clearly amenable to attack by the ratio test since it is straightforward to evaluate $\frac{a_{n+1}}{a_{n}}$. Whereas, for the second series, $\left(\frac{n}{n+1}\right)^{n^2}$, it is not immediately obvious how to evaluate $\frac{a_{n+1}}{a_{n}}$ in a workable form, and hence the ratio test is not easy or convenient to apply to it. In fact, since the terms contain $n$ in the power, this suggests that the root test will be helpful.

Experience of course helps a lot in deciding a lot on which test to use. It wouldn't hurt though to attempt to try several tests (there are definitely some series that can be tackled with multiple tests), if you are not immediately sure which one works.