AP Calculus BC glossary

Geometric series

A geometric series is one where each term is a fixed multiple of the previous term. It converges exactly when the absolute value of that common ratio is less than one, and then its sum is the first term divided by one minus the ratio.

\sum_{n=0}^{\infty} ar^n = \frac{a}{1-r}, \qquad |r| < 1

It is one of the very few series whose exact sum you can write down, which makes it the workhorse for comparison arguments and for building power series.

The mistake

Using the wrong first term. In the formula $a$ is the first term actually present in the series, so a sum starting at $n = 3$ has first term $ar^3$ and sums to $\frac{ar^3}{1-r}$ , not $\frac{a}{1-r}$ .

An annuity is a geometric series with a dollar sign. Each level payment is discounted by one more factor of $1/(1+r)$ than the one before it, so the future value of a stream of equal payments is a geometric sum: the future value of an annuity.

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