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

This is true or false :

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

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While it is nice to get questions from all over the world, the title of questions should be indicative of their content, not the location of the asker! You will surely agree tat this simple rule maximizes the utility of titles :) –  Mariano Suárez-Alvarez Jun 28 '12 at 18:11

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 $\sum_n a_n$).
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.