Homework: prove or disprove $\lim_{n\to\infty}n\sum_{k=n}^{\infty}b_k$ converge

Question

Suppose that $b_n>0$ for all $n$, that $$\sum_{n=1}^{\infty}b_n$$ and $$\sum_{n=1}^{\infty}nb_n$$ are all convergent. does that guarantee $$\lim_{n\to\infty}n\sum_{k=n}^{\infty}b_k$$ exists?

Answer 1

For $k \ge n$ note that $nb_k \le k b_k$, so that you have $$\sum_{k=n}^{\infty} nb_k \le \sum_{k=n}^{\infty} k b_k.$$ Since you have assumed that $\sum_{k=1}^{\infty} kb_k$ converges, use what you know about limits of tail ends of convergent series.