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When using the Ratio Test, having $$\limsup_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right| > 1$$ is inconclusive. However, I'm having trouble imagining how such a series $\sum a_n$ could possibly converge. Is it because when examining the sequence $\left\{\left|\dfrac{a_{n+1}}{a_n}\right|\right\}$, there is a subsequence that converges to something greater than one, while the rest of the sequence stays below one?

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what makes you think this situation is inconclusive? Besides, look at what happens when $a_n=2^n$. – Ittay Weiss Jan 19 '13 at 5:09
Usual Ratio Test is for series. And do you mean converge? – André Nicolas Jan 19 '13 at 5:10
Thanks for catching my mistakes, André! – angryavian Jan 19 '13 at 5:16
up vote 3 down vote accepted

Consider the series $$1+2+ r+2r +r^2+2r^2+r^3+2r^3+\cdots,$$ where (say) $r=\frac{1}{10}$. The series clearly converges, but $\limsup \left|\frac{a_{n+1}}{a_n}\right|=2$.

The ratio $a_{n+1}/a_n$ is "often" large, but it is also often small, and the small, in this example, beats the large.

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Your last sentence was a great explanation. Thanks! – angryavian Jan 19 '13 at 5:30

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