For each $n$ in $\mathbb{N}$, there exists $J_{n}$ finite set in $\mathbb{N}$ and $a_{i,n} \geq 0$ for $i \in J_{n}$ such that $\lim\limits_{n\to +\infty}\sum\limits_{i \in J_{n}} a_{i,n} = 0 $.
Suppose that there exists the limit $\lim\limits_{n\to +\infty}\sum\limits_{i \in J_{n}} i.a_{i,n}$. Can we say that $\lim\limits_{n\to +\infty}\sum\limits_{i \in J_{n}} i.a_{i,n} = 0$?
I think that it is not true. I found some questions related with it:
real analysis: $\sum a_n$ converges if and only if $\sum n a_{n^2}$ converges
Relate $\sum{a_n}$ and $\sum{n a_n}$
If $\sum n a_n$ converges, then $\sum a_n$ converges and $\sum |a_n|^p$ converges for $p>1$.
$J_n=\{n\}$, $a_{n,n}=\dfrac{1}{n}$.
$$\sum_\limits{i\in J_n}a_{i,n}=\frac1n\stackrel{n\to\infty}{\longrightarrow}0$$
But $$\sum_\limits{i\in J_n}ia_{i,n}=\frac n n\stackrel{n\to\infty}{\longrightarrow}1$$