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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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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