Let $\sum_{n=1}^\infty a_n$ be a convergent series. For every $n\in\mathbb{N}$, suppose that $\{a_{n,k}\}_{k=1}^\infty\subset[0,\infty)$ is a sequence that converges to $a_n\in[0,\infty)$ and that $\sum_{n=1}^\infty a_{n,k}$ is convergent.
Produce counterexamples to the following statements:
$\displaystyle{\lim_{k\to\infty}\sum_{n=1}^\infty a_{n,k}}=\sum_{n=1}^\infty a_n$
$\displaystyle{\limsup_{k\to\infty}\sum_{n=1}^\infty a_{n,k}}\leq\sum_{n=1}^\infty a_n$
$\displaystyle{\liminf_{k\to\infty}\sum_{n=1}^\infty a_{n,k}}\leq\sum_{n=1}^\infty a_n$
$a_n=0$ for all $n$, $a_{n,k}=1$ if $n=k$ and $0$ if $n \neq k$. This is a counter-example for all three.