$\textbf{Question}$： Let $\{z_n\}_{n=1}^{\infty}$ be any complex series while $\sum\limits_{n=1}^{\infty}\left|z_{n+1}^{-1}-z_n^{-1}\right|=\infty$, suppose $\sum\limits_{n=1}^{\infty}a_n z_n$ is convergent.$a_{n}$ is real.
Prove that： $$ \lim\limits_{N\rightarrow\infty}\left(\sum\limits_{n=1}^N a_n\right)\left(\sum\limits_{n=1}^N\left|z_{n+1}^{-1}-z_n^{-1}\right|\right)^{-1}=0. $$
$\textbf{Hint}$：
(1) May use $\textbf{Summation by Parts}$
(2) May use the $\textbf{Lemma}$：
Suppose $\varphi(n) \searrow 0(n \to \infty)$, and $\sum\limits_{n=1}^{\infty}a_{n}\varphi(n)$ is convergent,so we have: $$ \lim_{n \to \infty} \left( a_{1}+a_{2}+\cdots+a_{n} \right)\varphi(n)=0$$
$\textbf{My attempts and questions}$
First if $z_{n}$ is a constant sequence , so the result is trivial.
While $z_{n}$ is not a constant sequence. First by summation by parts I get $$ \left| \sum\limits_{n=1}^{N}a_{n}\right| = \left| \sum\limits_{n=1}^{N}a_{n}z_{n}z_{n}^{-1}\right| \leq M \left\{ \sum\limits_{n=1}^{N-1}\left|z_{n+1}^{-1}-z_{n}^{-1}\right| + \left|z_{N}^{-1}\right| \right\}, $$ where M = $\max\left\{ |a_{1}|,|a_{1}+a_{2}|,\cdots,|a_{1}+a_{2}+\cdots a_{N}| \right\}$
Next $$\left(\sum\limits_{n=1}^N a_n\right)\left(\sum\limits_{n=1}^N\left|z_{n+1}^{-1}-z_n^{-1}\right|\right)^{-1} \leq \dfrac{M \left\{ \sum\limits_{n=1}^{N-1}\left|z_{n+1}^{-1}-z_{n}^{-1}\right| + \left|z_{N}^{-1}\right| \right\}}{\sum\limits_{n=1}^N\left|z_{n+1}^{-1}-z_n^{-1}\right|} \leq M + \dfrac{M \left|z_{N}^{-1}\right| }{\sum\limits_{n=1}^N\left|z_{n+1}^{-1}-z_n^{-1}\right|}$$ Then I didn't know what to do next, I don't know the limitation of the rightmost term in the above equation, and I don't know how to estimate it.
