Can anyone prove that the ratio test is inconclusive when $\displaystyle \lim_{n \rightarrow \infty} |\frac{a_{n+1}}{a_{n}}| = 1,$ or $\displaystyle \lim_{n \rightarrow \infty} |\frac{a_{n+1}}{a_{n}}| DNE.$
Follow Math Help Forum on Facebook and Google+
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...
View Tag Cloud