# $u_n=\sum_{i=1}^{n} \frac{1}{i^\beta}$ converge [duplicate]

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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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
Hint: Let $U = u_{\infty}$. Think about the relationship between $iU$ and $U$.
What is for you exactly $\,u_\infty\,$?? –  DonAntonio Jan 7 '13 at 4:42