If $\sum\frac{a_{(n+1)}}{a_n}$ is convergent then can we say that $\sum a_n$ is also convergent?

Question

This is true or false :

If $\sum\frac{a_{n+1}}{a_n}$ is convergent then $\sum a_n$ is convergent.

Please help I have exam tomorrow!

Answer 1

If $\sum_n\frac{a_{n+1}}{a_n}$ is convergent, then in particular $\lim_{n\to +\infty}\frac{a_{n+1}}{a_n}=0$. Hence, for $n$ large enough (say $\geq n_0$), $|a_{n+1}|\leq \frac 12|a_n|$ so $|a_{n+n_0}|\leq \frac 1{2^n}|a_{n_0}|$ and we conclude that $\sum_n |a_n|$ is convergent (and so is $\sum_n a_n$).

Answer 2

Hint: If a series is convergent, the absolute value of the terms has limit $0$.

So if $n$ is large enough, $$\left|\frac{a_{n+1}}{a_n}\right| \lt \frac{1}{2},$$ that is, $$|a_{n+1}| \lt \frac{1}{2}|a_n|.$$ Now compare with a geometric series to conclude that in fact $\sum a_n$ is absolutely convergent.