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- Thread starter MissP.25_5
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The first series that you listed down, [itex]\frac{n!^{2}}{(2n)!}[/itex], is clearly amenable to attack by the ratio test since it is straightforward to evaluate [itex]\frac{a_{n+1}}{a_{n}}[/itex]. Whereas, for the second series, [itex]\left(\frac{n}{n+1}\right)^{n^2}[/itex], it is not immediately obvious how to evaluate [itex]\frac{a_{n+1}}{a_{n}}[/itex] in a workable form, and hence the ratio test is not easy or convenient to apply to it. In fact, since the terms contain [itex]n[/itex] 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.

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