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I'm having trouble with this statement:

"If $$\sum_{n=1}^\infty a_n$$ converges, and $a_n>0$ for all $n$, then $$\sum_{n=1}^\infty a_n^2$$ also converges"

I need to find either a proof or a counterexample. If somebody could give an idea, or a hint, it would be great.

Thanks so much!

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up vote 3 down vote accepted

If $\sum_{n=1}^{\infty} a_n $ converges, then this means, there is some $N$ such that $\forall n > N, a_n < 1$. Then, since $0 < a_n^2 < a_n, \forall n > N$, we have that $$\sum_{n=1}^{\infty} a_n^2 = \sum_{n=1}^{N} a_n^2 + \sum_{n=N+1}^{\infty} a_n^2 \le C + \sum_{n=N+1}^{\infty} a_n,$$ where $C<\infty$ is a constant, and $C = \sum_{n=1}^{N} a_n^2$. So, we have that $\sum_{n=1}^{\infty} a_n^2$ converges.

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Thank you so much! I´ll accept the answer as soon as they allow me to do so. – Lessa121 Feb 14 '14 at 5:42
I think a better exercise for you to think now is, what can you say about $a_n^i$ for $i > 0$? – Nameless Feb 14 '14 at 5:42
Well, I can apply the same principle, right? They all converge, even $a_n^n$ – Lessa121 Feb 14 '14 at 5:45
No problem. But, they don't quite all converge. The bigger $i$ is the faster they converge. (Assuming convergence of the original.) But, what happens when $i$ gets small? Consider the example $\sum_{n=1}^{\infty} \frac{1}{n^2}$. – mlg4080 Feb 14 '14 at 5:46
I was thinking of $i$ as an integer, but if it is in the interval [0,1], then not always, I guess. I could look for a counterexample, maybe $\frac 1 {n^2}$, $i=1/2$ – Lessa121 Feb 14 '14 at 5:48

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