Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of positive real numbers with finite Cesàro mean i.e., $$\lim\limits_{n\to\infty}\tfrac{1}{n}\sum_{i=1}^{n}a_i < \infty.$$ Prove or disprove $$ \lim\limits_{n\to\infty}\sum_{i=1}^{\infty}\frac{a_i}{n}e^{-\tfrac{i^2}{n}}<\infty. $$
I think it is not true. But I could not find a counterexample. Now for a fixed $n\in\mathbb{N}$, I was able to show $\sum_{i=1}^{\infty}\frac{a_i}{n}e^{-\tfrac{i^2}{n}}<\infty.$ For that I used the fact that since the Cesàro average is finite, $a_i \in o(i)$ and $\xi_n:=\sum_{i=1}^{\infty}\frac{i}{n}e^{-\tfrac{i^2}{n}}<\infty$.
Note that $a_i\in o(i)$, so we can find $c>0$ such that $a_i\leq c i$. Thus it suffices to show $\limsup_{n\to\infty} \xi_n<\infty$. Let $f_n(x)=\tfrac{|x|}{n}e^{-x^2/n}$ for $x\in\mathbb{R}$. It is easy to see, $$ \xi_n\leq \int_0^\infty f_n(x)\,dx. $$ Now \begin{align} \int_0^\infty f_n(x)\,dx &= \frac{1}{2}\sqrt{\frac{\pi}{n}}\int_{-\infty}^\infty \frac{|x|}{\sqrt{2\pi(n/2)}}e^{-\tfrac{x^2}{2(n/2)}}\,dx\\ &= \frac{1}{2}\sqrt{\frac{\pi}{n}}\,\mathbb{E}(|Z_n|)\\ &=\tfrac{1}{2}\sqrt{\tfrac{\pi}{n}}\sqrt{\tfrac{2}{\pi}}\sqrt{\tfrac{n}{2}}\\ &=\tfrac{1}{2} \end{align} where $Z_n\sim \mathcal{N}(0,\tfrac{n}{2})$. Thus $\limsup_{n\to\infty}\xi_n\leq\tfrac{1}{2}$ which proves the result.