# If a series is convergent, then does it have to pass the ratio test?

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?