# What are some examples of infinite geometric series?

Jul 13, 2015

Here are some examples:

$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots$

$1 - 1 + 1 - 1 + 1 - 1 + \ldots$

$1 + 2 + 4 + 8 + 16 + \ldots$

#### Explanation:

All geometric series are of the form ${\sum}_{i = 0}^{\infty} a {r}^{i}$ where $a$ is the initial term of the series and $r$ the ratio between consecutive terms.

In the three examples above, we have:

$a = 1$, $r = \frac{1}{2}$

${\sum}_{i = 0}^{\infty} a {r}^{i} = 2$

$a = 1$, $r = - 1$

${\sum}_{i = 0}^{\infty} a {r}^{i}$ does not converge - it alternates between $0$ and $1$ as each term is added.

$a = 1$, $r = 2$

${\sum}_{i = 0}^{n} a {r}^{i} \to \infty$ as $n \to \infty$

The geometric series ${\sum}_{i = 0}^{\infty} a {r}^{i}$ only converges in the following cases:

(1) $a = 0$

${\sum}_{i = 0}^{\infty} a {r}^{i} = 0$

(2) $\left\mid r \right\mid < 1$

${\sum}_{i = 0}^{\infty} a {r}^{i} = \frac{a}{1 - r}$