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

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


  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 $N\to\infty$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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