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Self-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges for $p > 1$

We know that $\lim u_n=\sum_{i=1}^{n} \frac{1}{i} =\infty $. But I can't prove that with $\beta >1$: $u_n=\sum_{i=1}^{n} \frac{1}{i^\beta}$ converge

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marked as duplicate by Marvis, Alexander Gruber, Ittay Weiss, Rahul, Austin Mohr Jan 7 '13 at 5:38

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Do you know the integral test? –  Brian M. Scott Jan 7 '13 at 4:31
Here is a beautiful argument by joriki. –  user17762 Jan 7 '13 at 4:31
thanks you so much –  Haruboy15 Jan 7 '13 at 4:33

1 Answer 1

Hint: Let $U = u_{\infty}$. Think about the relationship between $iU$ and $U$.

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What is for you exactly $\,u_\infty\,$?? –  DonAntonio Jan 7 '13 at 4:42

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