Can anyone prove that the ratio test is inconclusive when

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

$\displaystyle \lim_{n \rightarrow \infty} |\frac{a_{n+1}}{a_{n}}| DNE.$

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- Oct 12th 2009, 04:27 PMcgiulzProving the ratio test fails.
Can anyone prove that the ratio test is inconclusive when

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

$\displaystyle \lim_{n \rightarrow \infty} |\frac{a_{n+1}}{a_{n}}| DNE.$ - Oct 12th 2009, 04:31 PMJose27
For the first one take $\displaystyle a_n= \frac{1}{n}$ and $\displaystyle b_n= \frac{1}{n^2}$ both limits are 1, but one converges and the other diverges. For the second I'm not sure...