the definition of Riemann zeta function

Question

I've read that the Riemann zeta function for $0<s<1$ is defined :

$$\lim_{x\rightarrow\infty} \left(\sum_{n \leq x}\frac{1}{n^s}- \frac{x^{1-s}}{1-s}\right)$$

I don't know how to prove that this limit exists.

help me please.

thanks a lot.

Answer 1

Pg 55 of Apostol, theorem 3.2 (b) (halfway down page 56 he gives the proof) gives

$\sum_{n\leq x}\frac{1}{n^s}=\frac{x^{1-s}}{1-s} +\zeta(s) +O(x^{-s})$

if $1\neq s>0$ Subtract the first term of the right hand side and take $x$ to infinity.

(and the proof does consist of Euler Summation)