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I need to prove or disprove the following statement -

If $a_{n} >= 0 , p > 1$ , and $\sum_{i=1}^\infty{a_{n}}$ converge , then $\sum_{i=1}^\infty{a_{n}^p}$ converge

Well , It looks like a wrong statement but I couldn't think about familiar counterexample. I would like to get a hint.

Thanks!

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Hint: Prove that eventually $a_n < 1$, after which ${a_n}^p \le a_n$.

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The statement is true. Since $\sum a_n$ converges and $a_n\ge0$ we have $$\lim_n a_n=0.$$ This implies that exists $N>0$ such that $0\le a_n\le1$ for every $n>N$. Thus $$\sum_{n\ge N}a_n^p\le\sum_{n\ge N}a_n$$ and the statement is proved.

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  • $\begingroup$ I've no idea what this answer got downvoted whereas the other one got upvoted: they both say the same! $\endgroup$ – DonAntonio Apr 16 '16 at 11:35

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