AP Calculus BC glossary

p-series

A p-series is the sum of the reciprocals of the positive integers raised to a fixed power p. It converges when p is greater than one and diverges when p is less than or equal to one.

∑n=1∞1np converges  ⟺  p>1\sum_{n=1}^{\infty} \frac{1}{n^p} \ \text{converges} \iff p > 1

The rule follows from the integral test applied to 1xp\frac{1}{x^p}, which is why the boundary sits exactly at p=1p = 1: that is where the antiderivative switches from a power to a logarithm.

The mistake

Including p=1p = 1 as convergent. The harmonic series is the p=1p = 1 case and diverges, so the inequality is strict.

Appears in: Unit 10: Infinite Sequences and Series (BC)