Friday, December 9, 2011

The Professor, The Pretender and The Physicist.1

P-series

A generalization of the harmonic series is the p-series, defined as:

\sum_{n=1}^{\infty}\frac{1}{n^p},\!

for any positive real number p. When p = 1, the p-series is the harmonic series, which diverges. Either the integral test or the Cauchy condensation test shows that the p-series converges for all p > 1 (in which case it is called the over-harmonic series) and diverges for all p ≤ 1. If p > 1 then the sum of the p-series is ΞΆ(p), i.e., the Riemann zeta function evaluated at p.
link->Harmonic Series

p-Series Convergence

The p-series is given by
sum (1..inf) 1/np = 1/1p + 1/2p + 1/3p + ...
where p > 0 by definition.
If p > 1, then the series converges.
If 0 < p <= 1 then the series diverges.