$a_n > 0$ and $\sum_\limits{n=1}^{+\infty} \frac{1}{a_n}$ converges. Prove $\sum_\limits{n=1}^{+\infty} \frac{n}{a_1 + \cdots + a_n}$ is convergent.

Question

$a_n > 0$ and $\sum_\limits{n=1}^{+\infty} \frac{1}{a_n}$ converges. Prove $\sum_\limits{n=1}^{+\infty} \frac{n}{a_1 + \cdots + a_n}$ is convergent.

I find that this may have something to do with Stolz Theorem which says that if $\{\frac{1}{a_n}\}$ is convergent then $$\lim_{n \rightarrow +\infty} \frac{n}{a_1+\cdots +a_n} = \lim_{n \rightarrow +\infty} \frac{1}{a_n}$$ This may implies that $\frac{n}{a_1+\cdots +a_n}$ and $\frac{1}{a_n}$ have the same declining speed so leads to the answer of the question.

However, I don't know how to turn this into correct proof.

Answer 1

Let $b_n = \frac{1}{a_n}$, then $\frac{n}{a_1 + \cdots + a_n}$ is the harmonic mean of $b_1,..b_n$ so it is less or equal than the corresponding geometric mean.

$\sum_\limits{n=1}^{+\infty} \frac{n}{a_1 + \cdots + a_n} \leq \sum_\limits{n=1}^{+\infty} (b_1....b_n)^{\frac{1}{n}} \leq e\sum_\limits{n=1}^{+\infty} b_n=e\sum_\limits{n=1}^{+\infty} \frac{1}{a_n} < \infty$ by Carleman's Inequality