Upper bound for $\sum\limits_{k=N}^\infty \frac{1}{k^{1+\epsilon}}$

Question

Background: I'm trying to evaluate the Rieman Zeta function $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$$

I do not need a high precision, but I need to somehow estimate the error I'm making by cutting the sum of at some point.

So here my question:

Given a $\epsilon>0$ and an $N \in \mathbb N$ (e.g. $\epsilon < 2$) is there an upper bound for $$\sum_{n=N}^\infty \frac{1}{n^{1+\epsilon}} $$?

Answer 1

Hint. One may apply the integral test for convergence $$ \int_N^\infty f(x)\,dx\le\sum_{n=N}^\infty f(n)\le f(N)+\int_N^\infty f(x)\,dx $$ with $$ f(x):=\frac{1}{x^{1+\epsilon}}. $$