show if a sum is uniform convergent

Question

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

Answer 1

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