# Proving the ratio test fails.

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• October 12th 2009, 04:27 PM
cgiulz
Proving the ratio test fails.
Can anyone prove that the ratio test is inconclusive when

$\lim_{n \rightarrow \infty} |\frac{a_{n+1}}{a_{n}}| = 1,$ or

$\lim_{n \rightarrow \infty} |\frac{a_{n+1}}{a_{n}}| DNE.$
• October 12th 2009, 04:31 PM
Jose27
For the first one take $a_n= \frac{1}{n}$ and $b_n= \frac{1}{n^2}$ both limits are 1, but one converges and the other diverges. For the second I'm not sure...