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I've learned to apply these tests before in Calculus, but in the textbook that I used, the numbers of interest for the root and ratio tests were presented as $\lim_{n\rightarrow \infty}|a_n|^{1/n}$ and $\lim_{n\rightarrow \infty}a_{n+1}/a_n$ respectively. But now in Baby Rudin, we use instead the limit supremums and I am having a hard time understanding why. Are the tests made somehow more general using the limit supremum? Are there series which I wouldn't be able to apply the tests to without using the limit supremums?

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up vote 4 down vote accepted

Look at the series $$1+\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{36}+\frac{1}{72}+\cdots.$$ The ratio $\dfrac{a_{n+1}}{a_n}$ is alternately $\dfrac{1}{2}$ and $\dfrac{1}{3}$. So there is no such thing as $\lim_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|$.

However, $\limsup_{n\to\infty}\left|\dfrac{a_{n+1}}{a_n}\right|=\dfrac{1}{2}$, and we can conclude convergence.

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Thank you, this makes it clear why $\limsup$ makes for a better test. I have another, related question though (maybe I should make it a new question?): how do we KNOW that $\limsup_{n\rightarrow\infty}|a_{n+1}/a_n|=1/2$? I mean, I can see it and follow your argument, but how do I know that there isn't some subsequence, like, I don't know, the prime-numbered terms or something, which converge to something greater than 1/2? Is there a general method? – crf Nov 4 '12 at 6:10
$a_{n+1}/a_n$ is alternately $1/2$ and $1/3$. Subsequences are not relevant for the Ratio Test. If you want something less simple, we can modify things so that the ratios are alternately something that is different from $1/2$ and $1/3$, but that approach $1/2$ and $1/3$. One can make similar examples for the root test. – André Nicolas Nov 4 '12 at 6:28

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