CONTEXT: Question made up by uni lecturer
Say $\sum{a_n}$ is convergent. Does this then mean that $\lim_{n\to \infty}|\frac{a_{n+1}}{a_n}|\neq1$?
I know that the ratio test can be used to prove that a series converges, but I feel that there surely exists a convergent series where $\lim_{n\to \infty}|\frac{a_{n+1}}{a_n}|=1$ as it may have already passed another test proving its convergence.
Can anyone think of such a series?
TIA
Consider $\sum \dfrac1{n}$ and $\sum\dfrac 1{n^2}$. What does the ratio test give in each case?