TITLE: Prove that $\exists \{c_n\}$ monotonically increasing to $\infty$ such that $\sum_{i=1}^\infty a_nc_n$ coverges.
QUESTION [5 upvotes]: Given a real positive sequence $\{a_n\}$ such that $\sum_{i=1}^n a_i$ converges.
Prove that there exists a real sequence $\{c_n\}$ monotonically increasing to $\infty$ such that $\sum_{i=1}^\infty a_nc_n$ converges.
What I have done:
I have written
$$\sum_{i=1}^{N+1}a_ic_i=c_{N+1}\sum_{i=1}^{N+1}a_i-\sum_{i=1}^N(c_{n+1}-c_n)S_n$$
and then try to prove the convergence by the Cauchy criterion. However my evaluation is still not good enough. I'm really stuck.

REPLY [6 votes]: Put $\displaystyle R_n=\sum_{k\geq n} a_k$. Then $R_n>0$, and $R_n$ decrease to $0$ as $n\to +\infty$. We show that $\displaystyle c_n=\frac{1}{\sqrt{R_n}}$ (obviously increasing to $+\infty$) do the job.
We have:
$$a_n c_n=\frac{a_n}{\sqrt{R_n}}=\frac{R_n-R_{n+1}}{\sqrt{R_n}}\leq \int_{R_{n+1}}^{R_n}\frac{dx}{\sqrt{x}}$$
Hence for $N \geq 1$:
$$\sum_{n=1}^N a_n c_n \leq \int_{R_{N+1}}^{R_1}\frac{dx}{\sqrt{x}}\leq \int_0^{R_1}\frac{dx}{\sqrt{x}}<+\infty$$
Hence $\sum a_n c_n$ is convergent.