Convergence of P-Series/Real/Proof 1

Proof
By the Integral Test:


 * $\displaystyle \sum_{n \mathop = 1}^\infty \frac 1 {n^x}$ converges the improper integral $\displaystyle \int_1^\infty \frac {\d t} {t^x}$ exists.

The result follows from Integral to Infinity of Reciprocal of Power of x.