$ \sum_{n=1}^{\infty} n^{-z} $ converges uniformly

Question

Show that for every $ \delta > 0 $ the series

$$ \sum_{n=1}^{\infty} n^{-z} $$

converges uniformly on { $ z $ | $ Re(z) > 1 + \delta $ }

Conclude that the series defines in { $ z $ | $ Re(z) > 1 $ } the holomorphic function $\zeta(z) $

How to approach such thing? Bit confused :s

Answer 1

$|\frac 1 {n^{z}}|=\frac 1 {n^{x}}$ where $x =\Re z$. Since$\sum \frac 1 {n^a}<\infty$ for $a =1+\delta$ we can apply M-test.

Uniform convergence implies that the sum is anlaytic because it is the uniform limit of the partial sums which are analytic functions.