If $a_i\geq 0,$ prove $\sum\limits_{n=1}^\infty\frac{a_1+a_2+\cdots+a_n}{n}$diverges.
Solution 1:
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.