AP Calculus BC glossary

Harmonic series

The harmonic series is the sum of the reciprocals of the positive integers. Its terms shrink to zero, yet the series diverges, which makes it the standard proof that shrinking terms are not enough for convergence.

∑n=1∞1n=∞\sum_{n=1}^{\infty} \frac{1}{n} = \infty

It is the p-series with p=1p = 1, sitting exactly at the boundary between convergence and divergence, which is why it comes up constantly in comparison tests.

The alternating harmonic series does converge, to ln⁡2\ln 2. It is the classic example of conditional convergence: convergent as written, divergent once you take absolute values.

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