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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 $n>0$

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What did you try? What is your source? –  AD. Mar 28 '11 at 19:32
    
@Shai Covo: Note that this has a (homework) tag. It would be better to give a hint if anything at all - if I remember correctly it is also in Rudin's "Principles", as a problem. –  AD. Mar 28 '11 at 19:50
    
@AD. I deleted the reference to that book. –  Shai Covo Mar 28 '11 at 19:54
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@ Shai Covo: Great - then I delete my comment --- well no.. when I think about it it is best to leave these comments... –  AD. Mar 28 '11 at 19:59
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1 Answer 1

up vote 2 down vote accepted

Hint:

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.

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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. –  Chu Mar 28 '11 at 22:01
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