TITLE: Prove $\sum na_n$ converge if $\sum (a-s_n)$ converge
QUESTION [7 upvotes]: Let $\sum a_n=a$ with terms non-negative. Let $ s_n$ the n-nth partial sum.
Prove $\sum na_n$ converge if $\sum (a-s_n)$ converge

REPLY [3 votes]: $$
\begin{align}
\sum_{k=1}^nka_k
&=\sum_{k=1}^n\sum_{j=1}^ka_k\\
&=\sum_{j=1}^n\sum_{k=j}^na_k\\
&=\sum_{j=1}^n(s_n-s_{j-1})
\end{align}
$$
Taking limits, we get by Monotone Convergence that
$$
\begin{align}
\sum_{k=1}^\infty ka_k
&=\sum_{j=1}^\infty(a-s_{j-1})\\
&=a+\sum_{j=1}^\infty(a-s_j)\\
\end{align}
$$