Proof of p-series convergence criteria

$p$‍-series have the general form ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{\displaystyle \frac{1}{n^{{}^p}}}}$‍ where $p$‍ is any positive real number. They are convergent when $p>1$‍ and divergent when $0<p\le 1$‍.

The AP Calculus course doesn't require knowing the proof of this fact, but we believe that as long as a proof is accessible, there's always something to learn from it. In general, it's always good to require some kind of proof or justification for the theorems you learn.