Given $\lim_{n \rightarrow \infty} f_{n} = 0$ and $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} f_n = \infty$.
Suppose $\lim_{n \rightarrow \infty} \sum_{k=1}^{n} f_k \cdot g_k = M < \infty$.
[EDIT after Mark's reply]: Additional assumption of $g_n \geq 0$. What can we say about~$\lim_{n \rightarrow \infty} g_{n}$? Does the limit always exist? Is it always finite?
No, the limit may not exist, even if we impose $g_n>0$. Consider $f_n=1/n$ and $$ g_n=\begin{cases}1/n & \text{if }n\text{ is not a square,}\\ 1 & \text{if }n\text{ is a square.}\end{cases} $$ $$ \sum f_n\,g_n=\sum_{n\text{ not a square}}\frac1{n^2}+\sum_{k=1}^\infty\frac{1}{n^2}<\infty. $$