AP Calculus BC glossary

Divergent series

A series is divergent when its partial sums never reach a finite limit, either running off to infinity or oscillating forever without settling. Shrinking terms prevent neither: grouped into blocks that each total at least one half, the harmonic series has partial sums clearing every bound.

The harmonic series diverges even though its terms shrink to zero. Group the terms after the first as $\frac{1}{2}$ , then $\frac{1}{3} + \frac{1}{4}$ , then $\frac{1}{5}$ through $\frac{1}{8}$ , and so on. Every block totals at least $\frac{1}{2}$ , so the partial sums pass any bound you name, however slowly.

Two different failures both count as divergence. The partial sums of $\sum_{n=1}^{\infty} \frac{1}{n}$ climb past every bound, while the partial sums of $\sum_{n=1}^{\infty} (-1)^n$ bounce between $-1$ and $0$ forever. Saying a series diverges to infinity therefore claims more than saying it diverges.

The mistake

Feeding a divergent geometric series into the sum formula anyway. With $a = 1$ and $r = 2$ , the expression $\frac{a}{1-r}$ returns $-1$ for a series of positive growing terms. Confirm $|r| < 1$ before the formula means anything at all.

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