Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Help please

Let $(a_{n})_{n>o}$ and $(b_n)_{n>o}$ be sequences of complex numbers and suppose that:

(i) The sequence of partial sums $S_{m}=\sum_{n=0}^{m}a_{n}$ is bounded

(ii) $\mathrm{lim}b_{n}=0$

(iii) The sum $\sum_{n=1}^{\infty} |b_{n} - b_{n-1}|$ is finite.

Prove that the series $\sum a_{n}b_{n}$ is convergent. Hint: Use "Abel summation":

$\sum_{k=n}^{m}a_{k}b_{k}=\sum_{k=n}^{m}(S_{k} - S_{k-1})b_{k}$ for $n>0$

share|cite|improve this question
What did you try? What is your source? – AD. Mar 28 '11 at 19:32
up vote 2 down vote accepted


1) Put $C_n=\sum_{k=0}^na_kb_k$, and note that is is sufficient to show that $C_n$ is a Cauchy sequence.

2) Now use the hint given in the problem: $$C_m-C_{n-1} = \sum_{k=n}^{m} (S_k-S_{k-1})b_k$$ Break up the last sum in two sums and change the index in of of those sums in order to apply what you know about $(b_k)$.

I hope this helps - please ask if there is some problem.

share|cite|improve this answer
I was able to do it this way. I think I should say however that one has to prove after your hint that the resulying series converges absolutely. This is the step that I was struggling with. Thank you for your answer. It helped me a lot. – Sak Mar 28 '11 at 22:01

Your Answer


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.