quality:correction, structural:scholar_edition, information:filtering, information:classification
p -series test

A commonly-used corollary of the integral test is the p-series test. Let k > 0 {\displaystyle k>0}. Then ∑ n = k ∞ (1 n p) {\displaystyle \sum _{n=k}^{\infty }{\bigg (}{\frac {1}{n^{p}}}{\bigg)}} converges if p > 1 {\displaystyle p>1}.

The case of p = 1, k = 1 {\displaystyle p=1,k=1} yields the harmonic series, which diverges. The case of p = 2, k = 1 {\displaystyle p=2,k=1} is the Basel problem and the series converges to π 2 6 {\displaystyle {\frac {\pi ^{2}}{6}}}. In general, for p > 1, k = 1 {\displaystyle p>1,k=1}, the series is equal to the Riemann zeta function applied to p {\displaystyle p}, that is ζ (p) {\displaystyle \zeta (p)}.