Convergence of P-Series/Divergence if p between 0 and 1

Theorem
Let $p$ be a complex number. Let $0 < \map \Re p \le 1$.

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

diverges.

Lemma
As proved above, the convergence of the $p$-series is dependent on the convergence of:


 * $\displaystyle \lim_{t \mathop \to \infty} \frac{t^{1 - x}} {1 - x}$

If $x = 1$, the series clearly diverges because of Division by Zero.

Suppose $0 < x < 1$.

Then:

Again, the result follows from the Integral Test.