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Prove that if $\sum a_n$ is a convergent series of nonnegative numbers and $p>1$, then $\sum a_n^p$,converges.

My proof is as follows.

By the theorem which says that if a series $\sum a_n$ converges, then $\lim_{n \rightarrow \infty} = 0$, the sequence $(a_n)$ approaches 0 as n approaches infinity. This implies each $a_n$ is in the form of fraction where the denomenator is greater than numerator. Then, $|a_n^p| \leq a_n$, and $\sum a_n^p$ converges by the comparison test.

Is this valid?? I saw back of my text book and gives different proof!

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up vote 1 down vote accepted

You say $a_n$ is " the form of fraction where the denominator is greater than numerator." Your idea is fine, but what you want to say is that $a_n<1$ eventually since $a_n\to 0$, so that $$a_n^{1+\varepsilon}=a_n^{\varepsilon}a_n<a_n$$

for in the interval $0<x<1$, we have $x^{\varepsilon }<1$

Note the absolute value bars are unnecessary, since $a_n\geqslant 0$.

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