# How to prove the following series converges?

The $\sum \limits_{n=0}^{\infty} \dfrac {a_n}{n}$ where $a_n$ is any sequence that meets the condition: $\lim \limits_{n \to \infty} a_n=0$.

I have tried to apply the different tests available such as the limit comparison test and divergence test but I am not sure how to rigorously prove this statement. Any help would be appreciated.

• The statement is false. – user296602 Mar 10 '16 at 5:21
• I heavily edited your question. Please be sure I did not change its meaning. – zz20s Mar 10 '16 at 5:23
• The answer you accepted does not answer your edited question. – fosho Mar 10 '16 at 6:04
• The sum should start at $n=1$ since $\frac{a_n}n$ is bad for $n=0$. – robjohn Mar 10 '16 at 6:10

If $$a_n=\frac1{\log(n+1)}$$ then the series $$\sum_{n=1}^\infty\frac{a_n}n$$ does not converge.
• Apply the Cauchy Condensation Test: If $f(n) \geq f(n+1)\geq 0$ for all but finitely many $n$ then $\sum_{n=1}^{\infty} f(n)$ converges iff $\sum_{n=1}^{\infty}[2^n f(2^n)]$ converges. – DanielWainfleet Mar 10 '16 at 7:29
This not true, see $a_n = \frac{1}{\sqrt{n}}$.
• Is this supposed to be a counterexample? The series $\sum n^{-3/2}$ converges. – RRL Mar 10 '16 at 6:25