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I want to ask the following problem:

Let $a_n$ be a sequence of positive numbers such that $\sum a^2_n$ converges. For $b_n = \dfrac{a_1 + a_2 + \cdots + a_n }{n}$, does $\sum b^2_n$ converges?

At first, I tried to find a counterexample by computing the case where $a_n = 1/n^{1/2 + \epsilon} $, but sadly that made $\sum b^2_n$ convergent. Also, I tried to prove that the sequence converges by using C-S inequality, limit comparison test, etc... but all ideas failed.

Please give me your advice or ideas for this problem. Thank you in advance.

Peter James
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1 Answers1

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Try applying Hardy's inequality. Your case is $p = 2$ and you have $\sum b_n^2 \leq 4 \sum a_n^2$, but you already know that $\sum a_n^2$ converges. Can you work with the definition of series convergence from here?

eordo
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  • Yes, I found that the leftover is straightforward. Thanks a lot! I'll work myself in order to prove the hardy's inequality. (not using the integrated version) – Peter James Feb 19 '21 at 17:42