Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
    
Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. –  Cameron Buie Nov 14 '12 at 1:49
add comment

1 Answer 1

up vote 4 down vote accepted

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.

share|improve this answer
    
Aha, I see. Thank you! –  asdgdffasd Nov 14 '12 at 1:42
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.