# Convergence of a complex product series

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

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 $(b_k)$.