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Infinite geometric series

An infinite geometric series has a finite sum only when the absolute value of the ratio is less than 1. Then the terms become smaller and smaller, and the sum is the first term divided by one minus the ratio. If the absolute value of the ratio is greater than or equal to 1, the terms grow or stay the same size, and the sum does not exist as a number. You must therefore check the ratio before you use the formula.

Try it yourself

Geometric series

S = \frac{a_1}{1 - k}

sum when |k| < 1

|k| < 1

condition for convergence

Symbols

$a_1$	the first term
$k$	the ratio
$S$	the sum

Example

With $a_1 = 6$ and $k = \dfrac{1}{3}$ , $|k| < 1$ , and $S = \dfrac{6}{1 - 1/3} = 9$ . The series $1 + \dfrac{1}{2} + \dfrac{1}{4} + \cdots$ has sum $2$ .

The formula does not apply for

k = 2

. Then the terms grow, and the sum increases without bound.

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Part of Mathematics R2: Sequences and series.