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?
Homework: prove or disprove $\lim_{n\to\infty}n\sum_{k=n}^{\infty}b_k$ converge
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calculus
sequences-and-series
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1 Answers
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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.
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0Aha, I see. Thank you! – 2012-11-14