AP Calculus BC glossary

Sum of a geometric series

Also called: Geometric series sum

For a geometric series whose common ratio has absolute value less than one, the sum equals the first term present divided by one minus the ratio. This is one of the very few infinite series with a clean closed-form total, and it powers many comparison and power-series arguments.

The aa in a1−r\frac{a}{1-r} is the first term that actually appears. So ∑n=2∞3⋅(12)n\sum_{n=2}^{\infty} 3 \cdot \left(\frac{1}{2}\right)^n has first term 3⋅14=343 \cdot \frac{1}{4} = \frac{3}{4} and sums to 341−12=32\frac{\frac{3}{4}}{1 - \frac{1}{2}} = \frac{3}{2}.

The closed form comes from the partial sum SN=a1−rN1−rS_N = a\frac{1 - r^N}{1 - r}. When ∣r∣<1|r| < 1, letting N→∞N \to \infty sends rNr^N to zero, leaving a1−r\frac{a}{1-r}, which is why the size of rr decides everything.

The mistake

Using the formula when it does not apply. The closed form a1−r\frac{a}{1-r} holds only for ∣r∣<1|r| < 1. If ∣r∣≥1|r| \ge 1 the series diverges and has no sum, so plugging in and reporting a finite number is simply wrong.

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