Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I've got a question about the convergence of a series during studying analysis.

If I know that a series of positive real numbers $$\sum_{n=1}^\infty a_n$$ converge, why does $$\sum_{n=1}^\infty\left(\frac{a_n}{n^p}\right)^\frac{1}{2}$$ also converge for $p>1$?

Although I know about many convergence tests, I don't know how to apply those tests for this case. Since this problem is the form of "series A converge → series B converge", I've been thinking that it must be verified by using some "comparison" tests. Is this thinking correct?

All advice is welcome^_^

share|improve this question
@Landscape : Yes. I missed it. I'll update now. –  Analysis Jun 13 '13 at 3:08
@Maesumi : But then, $\sum1$ does not converge which is not fulfilled the condition. –  Analysis Jun 13 '13 at 3:49

2 Answers 2

up vote 5 down vote accepted

Since $$ab\leq a^2+b^2$$ we have $$\sum_{n=1}^\infty\left(\frac{a_n}{n^p}\right)^\frac{1}{2}\leq \sum_{n=1}^\infty a_n+\sum_{n=1}^\infty\frac{1}{n^p} $$

share|improve this answer
@landscape Why you deleted your answer? As I have not deleted my answer (and I do not do) you must not remove yours, that's what you said. –  Sami Ben Romdhane Jun 13 '13 at 3:31
Dear Sami Ben Romdhane, I deleted mine because I thought it was a little embarrassing that my answer posted later than yours, but shared the same idea with yours and contained less details than yours. If you don't mind, I will undelete it. –  23rd Jun 13 '13 at 3:39
@Landscape A hint shouldn't contains more detail so I'd be happy if you undelete it. –  Sami Ben Romdhane Jun 13 '13 at 3:46
I have undeleted my answer. Thank you. –  23rd Jun 13 '13 at 3:53
Nice work, as usual! –  amWhy May 31 '14 at 11:24

Hint: $$\left(\frac{a_n}{n^p}\right)^{\frac{1}{2}}\le\frac{1}{2}\left(a_n+\frac{1}{n^p}\right).$$

share|improve this answer
Wow. Really simple! How do you think this inequality... Thanks! –  Analysis Jun 13 '13 at 3:11
@Analysis: You are welcome. Since Sami Ben Romdhane posted a full answer a little earlier than me, let me delete my answer later. –  23rd Jun 13 '13 at 3:14
@Analysis: I am sorry for being capricious. After some discussion with Sami Ben Romdhane, I undeleted my answer. –  23rd Jun 13 '13 at 3:57

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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