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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 get smaller and smaller, and the sum is the first term divided by 1 minus the ratio. If the absolute value is greater than or equal to 1, the terms grow or stay the same size, and the sum does not exist as a number. 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

the condition

Symbols

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

Example

$a_1 = 12$ and $k = 0.25$ give $S = \dfrac{12}{1 - 0.25} = 16$ .

The series $1 + \dfrac{1}{2} + \dfrac{1}{4} + \cdots$ has sum $\dfrac{1}{1 - 0.5} = 2$ .

The formula does not apply when the ratio is 1 or greater.

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Part of Mathematics S2: Sequences and series.