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

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 $a_i\neq 0$.

• The $n$-th partial sum is clearly greater than $\sum_{i=1}^n a_1/i = a_1\sum_{i=1}^n 1/i$.
– Tad
Apr 5 '15 at 15:43
• How do you prove $\sum\limits_{n=1}^\infty \sqrt[n]{a_1a_2\cdots a_n}$ and $\sum\limits_{n=1}^\infty\frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}}$ converge? Thanks.
– Hans
Apr 9 '15 at 19:19
• @Hans,The provement may be long and Will I type them here? Apr 10 '15 at 14:29
• I have opened a question math.stackexchange.com/questions/1228749/…. You may post your answer there.
– Hans
Apr 10 '15 at 18:54

## 1 Answer

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.

• What is $\;a_{n_0}\;$ and how come that mean is greater than $\;a_{n_0}\cdot\frac1n\;$ ?? Apr 5 '15 at 15:45
• Why is $a_i \ge a_{n_0}$?
– Keba
Apr 5 '15 at 15:46
• Just discard all terms except $a_{n_0}$, and since they are positive... Apr 5 '15 at 15:47
• Ah, my fault. I totally misread your statement.
– Keba
Apr 5 '15 at 15:50
• Wow, after seeing this answer I feel incredibly stupid. Apr 5 '15 at 15:59