Lesson 4: Finding Sums of Infinite Geometric Series

Property

For an infinite geometric series whose first term is $a_1$ and common ratio $r$,

If $|r| < 1$, the sum is

Property

To convert a repeating decimal to a geometric series:

1. Identify the repeating block and its position
2. Express as: non-repeating part + $\frac{\text{repeating block}}{10^k} + \frac{\text{repeating block}}{10^{2k}} + \frac{\text{repeating block}}{10^{3k}} + \ldots$
3. The repeating portion forms a geometric series with $a_1 = \frac{\text{repeating block}}{10^k}$ and $r = \frac{1}{10^k}$

Examples

Property

A repeating decimal can be converted to a fraction by expressing it as an infinite geometric series. For a repeating decimal $0.\overline{d_1d_2...d_n}$, we can write it as a geometric series and use the formula $S = \frac{a}{1-r}$ where $|r| < 1$.

Examples