TITLE: If $a_i\geq 0,$ prove $\sum\limits_{n=1}^\infty\frac{a_1+a_2+\cdots+a_n}{n}$diverges.
QUESTION [10 upvotes]: If $a_i\geq0,a_n\not\equiv 0,$ prove  $\sum\limits_{n=1}^\infty\frac{a_1+a_2+\cdots+a_n}{n}$ diverges.
I have known that if $\sum\limits_{n=1}^\infty a_n$converges, then  $\sum\limits_{n=1}^\infty \sqrt[n]{a_1a_2\cdots a_n}$and $\sum\limits_{n=1}^\infty\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}}$converges, but how can I prove $\sum\limits_{n=1}^\infty\frac{a_1+a_2+\cdots+a_n}{n}$ diverges?
Sincerely thanks for your help.


Add : Such stupid as I am, we do not need the condition $\sum a_n$ converges. 


Strictly, rewrite $\sum\limits_{n=1}^\infty\frac{a_1+a_2+\cdots+a_n}{n}=\sum\limits_{n=k}^\infty\frac{a_1+a_2+\cdots+a_n}{n}\geq a_k\sum\limits_{n=k}^\infty\frac 1n\to 
\infty$, where $a_k$ is the first $a_i\neq 0$.

REPLY [13 votes]: Suppose $a_{n_0}\neq 0$, then for all $n\geq n_0$, 
$$\frac{a_1+\cdots+a_n}{n}\geq a_{n_0}\times\frac{1}{n}$$
hence diverges since the harmonic series diverges.