Whether a series is convergent or divergent Is it true that if $\sum u_n$ is convergent, where $u_n$'s are positive real numbers then $\sum \dfrac{u_1+u_2+...+u_n}{n}$ is divergent?
I know that if   $\lim_{n\to\infty}u_n =0$ then $\lim_{n\to\infty}\dfrac{u_1+u_2+...+u_n}{n}=0$ and it is the necessary condition for a series to be convergent.
Someone help please.

The post has been edited to say the $u_i$'s are positive.  Under that assumption, $\sum \dfrac{u_1+u_2+...+u_n}{n}$ diverges because $\sum_{n=1}^m \dfrac{u_1+u_2+...+u_n}{n}>\sum_{n=1}^m \dfrac{u_1}{n}$ and the latter goes to infinity as $m\to\infty$.
