Geometric Series

Summary

A geometric series has a constant ratio between consecutive terms. It converges precisely when $| r | < 1$ , with sum $a / (1 - r)$ .

Prerequisites

Series Sums by Partial Sums

Definition

\sum_{n = 0}^{\infty} a r^{n} = a + a r + a r^{2} + \dots .

Theorem

If $| r | < 1$ , the series converges to $\frac{a}{1 - r}$ .
If $| r | \geq 1$ and $a \neq 0$ , the series diverges.

Special cases: $r = 1$ gives $a + a + a + \dots$ ; $r = - 1$ oscillates; $r = 0$ is the trivial one-term series $a$ .

Worked Example

With $a = 2$ , $r = 1 / 2$ :

2 + 1 + \frac{1}{2} + \frac{1}{4} + \dots = \frac{2}{1 - 1 / 2} = 4.

Common Mistakes

Using $a / (1 - r)$ when $| r | \geq 1$ .
Off-by-one errors in the starting index (sum from $n = 1$ vs $n = 0$ ).

Connections

Power Series, Ratio Test, remainder formulas in Estimating Series Sums

References

Geometric series are foundational in OpenStax Calculus Volume 2.^[1]

OpenStax, Calculus Volume 2, Section 5.2, https://openstax.org/details/books/calculus-volume-2 ↩︎