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Let $(x_n)$ and $(y_n)$ be sequences such that the sequence of partial sums of the series $\sum_{n=1}^\infty{x_n}$ is bounded, and furthermore the series

$$\sum_{n=1}^\infty{|y_n - y_{n+1}|} $$ converges, while $y_n \to 0 $ for $x \to \infty$

Prove that the series $\sum_{n=1}^\infty{x_n{y_n}}$ converges.

Hint: use the summation by parts formula

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Are you familiar with the Abel transform or, alternatively, do you know how to rewrite $\sum x_ny_n$ using the differences $|y_n-y_{n-1}|?$ – leshik Sep 30 '12 at 23:25
up vote 2 down vote accepted


  1. Check that $$ \sum\limits_{n=1}^N x_ny_n= y_N X_n-\sum\limits_{n=1}^N X_n(y_{n+1}-y_n) $$ where $X_n=\sum\limits_{n=1}^N x_n$

  2. Since $X_n$ is bounded $$ \sum\limits_{n=1}^N x_ny_n\leq M(|y_N|+\sum\limits_{n=1}^N|y_{n+1}-y_n|) $$ for some $M\geq 0$

  3. It is remains to take a limit $N\to\infty$.

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