convergence of Riemann zeta series

The series

${\displaystyle \sum _{n=1}^{\mathrm{\infty }}}{\displaystyle \frac{1}{n^s}}$ (1)

converges absolutely for all $s$ with real part greater than 1.

Proof. Let  $s=\sigma +it$  where  $\sigma$ and $t$ are real numbers and  $\sigma >1$.  Then

$$\left|\frac{1}{n^s}\right|=\frac{1}{|e^{s\log n}|}=\frac{1}{e^{\sigma \log n}}=\frac{1}{n^{\sigma }}.$$

Since the series  ${\sum }_{n=1}^{\mathrm{\infty }}\frac{1}{n^{\sigma }}$ converges, by the $p$-test (http://planetmath.org/PTest), for  $\sigma >1$, we conclude that the series (1) is absolutely convergent in the half-plane  $\sigma >1$.

Title	convergence of Riemann zeta series
Canonical name	ConvergenceOfRiemannZetaSeries
Date of creation	2015-08-22 13:15:14
Last modified on	2015-08-22 13:15:14
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	8
Author	pahio (2872)
Entry type	Definition
Classification	msc 11M06
Classification	msc 30A99
Related topic	ModulusOfComplexNumber
Related topic	ComplexExponentialFunction