Convergence of a product series with one $1/k$ factor

Question

Let $\left( a_n \right)_n$ be a sequence such that $a_n < 1$, $a_n \rightarrow 0$.

Prove or disprove (with a counter-example) that $$ \sum_{n=1}^{\infty} \frac{a_n}{n} < \infty.$$

Comments. If there exists $c, e>0$ such that $a_n \leq c/n^e$ for all $n$, than we get the upper bound $c \sum_n 1/n^{1+e} < \infty$. But I am not sure that such $c, e$ always exist. Does $a_n = 1/\log(n)$ disprove the convergence?

Answer 1

An idea:

Take $\;a_n:=\dfrac1{\log n}\;,\;\;n\ge3$ .

You might also want to use the Condensation Test to check stuff with the resulting series.

Answer 2

Use integral test, since $1/n\ln n\to 0$ and is decreasing, the convergence of $\sum 1/n\ln n$ is the same as $\int_{2}^{+\infty} dx/x\ln x$...