# Ratio test and root test

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.

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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.