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

Possible Duplicate:
Series converges implies $\lim{n a_n} = 0$

Someone can help me? If $(a_n)$ is a decreasing sequence and $\sum a_n$ converges. Then $\lim {(n.a_n)} = 0$.

I don't have idea how to solve this.

share|cite|improve this question

marked as duplicate by Martin Sleziak, Thomas, rschwieb, Noah Snyder, Emily Oct 22 '12 at 18:27

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

By the Cauchy condensation test

$$\sum 2^m a_{2^m} < \infty$$


$$\lim_n 2^m a_{2^m} =0$$

Now, for each $n$ chose some $m$ so that $2^m \leq n < 2^{m+1}$ and use

$$a_{2^m} \geq a_n \geq a_{2^{m+1}}$$

share|cite|improve this answer

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