Ratio test - Ratio tends to infinity
Matthew Harrington 
I have to prove that if $a_n\neq0$ for all n, and $|\frac{a_{n+1}}{a_n}|\to \infty$, then $\sum a_n$ diverges.
I have proved the ratio test for when the limit of the ratio, L, is less than 1 or greater than 1. But I think I have to be a bit more careful here, rather than going straight from the previous result.
Also, when a question says 'if the limit exists', is infinity included in this? Thanks.
$\endgroup$2 Answers
$\begingroup$Hint There exists an $N$ so that for all $n >N$ we have
$$|\frac{a_{n+1}}{a_n}| >2$$
Thus $$|a_{n+1}|>2|a_n|$$
It follows that $|a_{n+1}|$ is increasing from $N$ and positive. It follows that $|a_n|$ cannot converge to $0$, and hence $a_n$ doesn't converge to $0$.
$\endgroup$ $\begingroup$Note that for large $n$ we can have $\frac{|a_{n+N}|}{|a_n|} > 1$ for any $N > 1$ , so we can use this to deduce that $\lim_{N \rightarrow \infty} |a_{n+N}| \neq 0 $ , so the series cannot converge.
For the second question, the limit exist depends on the context you are working with, some textbooks include $\infty$ and others not. But I consider this not a great problem , always it is good to try to see what happens when limit is $\infty$ , where it may gives us intuition as for what is happening when objects get very large .
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