If $\sum a^2_n$ converges so does $\sum \frac{a_n}{n}$
I'm trying to prove/ disprove this statement. But I couldn't end up with anything yet. Any help?
If $\sum a^2_n$ converges so does $\sum \frac{a_n}{n}$
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real-analysis
sequences-and-series
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0(in particular, through the fact that $\sum_{n\in\Bbb N} \frac{\lvert a_n\rvert}n$ converges). – 2017-02-23
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0See also [here](http://math.stackexchange.com/questions/2056157/show-that-if-sum-n-1-inftya-n2-converges-then-sum-n-1-inftya-n?noredirect=1&lq=1). – 2017-02-23
1 Answers
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Hint: Using the following inequallity $(a_n-\frac{1}{n})^2\ge 0$ one has $$ a^2_n+\frac{1}{n^2}\ge 2 a_n \frac{1}{n}$$. Dividing by 2 , taking sums you have the convergence of the greater sum, so your series converges