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my teacher recommended to use this if I need hints on homework help. I am trying to determine that the series $\sum_{n=1}^\infty \frac{\xi}{n}$ on $\xi \in (0,1)$ converges uniformly or doesn't converge at all. I'm going to guess that it does not converge uniformly. I had an idea to show that it was not a cauchy uniform sequence then it would work. I denote the partial sums by $f_n$ and consider $||f_{n} - f_{n-1}||_{\sup} = ||\frac{\xi}{n}||_{\sup} = 1/n$ so this leads to no results. Could someone link me to the right directions

toothandcup2
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1 Answers1

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The sum $$ \sum_{n=1}^{\infty}\frac{\zeta}{n}=\zeta\sum_{n=1}^{\infty}\frac{1}{n} $$

diverges since $\sum_{n=1}^{\infty}\frac{1}{n}$ does.

The sum $\sum_{n=1}^{\infty}\frac{1}{n}$ is called the harmonic series

Belgi
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  • but $\xi$ is not a constant? is a variable between $0,1$ – toothandcup2 Mar 07 '15 at 17:41
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    For the purposes of each sum, it's a constant. – Alan Mar 07 '15 at 17:42
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    @toothandcup2 Do not confuse what is varying. In the sum, the only thing that is changin is $n$; so anything that depends on $n$ is changing, anything that does not depend on $n$ can be treated as a constant. $\zeta$ may be changing outside the sum, but it's irrelevant – Ant Mar 07 '15 at 17:43
  • Then I am confused, because in lecture we proved that if $\xi \in (0,m) $ for $0 < m < 1$ then the sum converges uniformly, now I am trying to show it diverges. Could I not use the same argument as you used to show that for $\xi \in (0,m)$ diverges? – toothandcup2 Mar 07 '15 at 17:45
  • @Alan does this mean the lecture was wrong? – toothandcup2 Mar 07 '15 at 17:53
  • @toothandcup2 - that statement doesn't sound right - but probably you are misinterpreting what the lecture stated – Belgi Mar 07 '15 at 18:01
  • He said$ \sum_n \xi / n $ converges uniformly for $\xi \in (0,m)$ for $0<m<1$ then then asked us if it converges uniformly for $\xi \in (0,1)$ – toothandcup2 Mar 07 '15 at 18:03