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 $a$ in $\frac{a}{1-r}$ is the first term that actually appears. So $\sum_{n=2}^{\infty} 3 \cdot \left(\frac{1}{2}\right)^n$ has first term $3 \cdot \frac{1}{4} = \frac{3}{4}$ and sums to $\frac{\frac{3}{4}}{1 - \frac{1}{2}} = \frac{3}{2}$ .

The closed form comes from the partial sum $S_N = a\frac{1 - r^N}{1 - r}$ . When $|r| < 1$ , letting $N \to \infty$ sends $r^N$ to zero, leaving $\frac{a}{1-r}$ , which is why the size of $r$ decides everything.

The mistake

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

This sum is where the economic multiplier comes from. Each round of spending is the previous round times the marginal propensity to consume, so total spending is a geometric series and the familiar $1/(1-\text{MPC})$ is nothing more than $a/(1-r)$ applied to it: the spending multiplier.

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