Convergence of P-Series/Absolute Convergence if Real Part of p Greater than 1

Theorem
Let $p$ be a complex number. Let $\map \Re p > 1$.

Then the $p$-series:
 * $\displaystyle \sum_{n \mathop = 1}^\infty n^{-p}$

converges absolutely.

Lemma
Since $x > 1$ it follows that $1 - x < 0$.

Thus $P^{1 - x} \to 0$ as $P \to \infty$.

Setting $x - 1 = \delta >0$, this limit is:


 * $\displaystyle -\frac 1 {\delta} \lim_{t \mathop \to \infty} \frac 1 {t^\delta} = 0$

Hence the integral is just $\dfrac 1 {1 - x}$ (that is, convergent) and so the sum converges as well.

Since the terms of the sum were positive everywhere, it is absolutely convergent and hence so is:


 * $\displaystyle \sum_{n \mathop = 1}^\infty n^{-p}$