TITLE: Convergence criteria for real Riemann zeta function
QUESTION [5 upvotes]: It is well known that the following series converges for $x > 1$, where $x$ is real.
$$ \zeta(x) = \sum_{n=1}^{\infty} \frac{1}{n^x}$$
Why is this? I have searched and can't find an answer, and most sources read as if it is so obvious it doesn't need an explanation.
For $x=1$ I know this is the divergent harmonic series.
A ratio test fails to conclude convergence:
$$ \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{(n+1)^x}{n^x} = 1$$
I've even plotted a graph of this and observed that as $x$ varies between 0 and 10, the limit still approaches 1.
I'm not mathematically trained so I'd appreciate explanations with minimum technical language.

REPLY [3 votes]: I don't know why you think that Cauchy Condensation involves complex analysis.


CC Suppose $a_n\ge0$ and $a_{n+1}\le a_n$. The series $\sum a_n$ converges if and only if the series $\sum 2^na_{2^n}$ converges.


No complex numbers in sight. 
Now suppose $x>0$ and $a_n=1/n^x$. Then $$2^na_{2^n}=2^{(1-x)n},$$and it's clear that $\sum 2^{(1-x)n}$ converges if and only if $1-x<0$.