# How do you find the sum of the infinite geometric series -5 – 5/2 – 5/4 - . . .?

Jun 18, 2018

$\left(- 5\right) + \left(- \frac{5}{2}\right) + \left(- \frac{5}{4}\right) + \left(- \frac{5}{8}\right) + \ldots = \textcolor{b l u e}{- 10}$

#### Explanation:

Consider the geometric series defined as
$\textcolor{w h i t e}{\text{XXX}} {a}_{n} = \frac{1}{{2}^{n}}$

${\Sigma}_{i = 0}^{\infty} {a}_{i} = 2$
$\textcolor{w h i t e}{\text{XXXXXXXXXXX}}$[see below, if necessary for why this is true]

Each term of the given geometric series is simply $\left(- 5\right)$ time the corresponding term of ${a}_{n}$.
Therefore $\Sigma \left(\left(- 5\right) + \left(- \frac{5}{2}\right) + \left(- \frac{5}{4}\right) + \ldots\right)$

$\textcolor{w h i t e}{\text{XXXXXX}} = {\Sigma}_{i = 0}^{\infty} \left(- 5\right) \cdot \left({a}_{i}\right)$

$\textcolor{w h i t e}{\text{XXXXXX}} = \left(- 5\right) \cdot {\Sigma}_{i = 0}^{\infty} {a}_{i}$

$\textcolor{w h i t e}{\text{XXXXXX}} = \left(- 5\right) \cdot 2$

$\textcolor{w h i t e}{\text{XXXXXX}} = - 10$

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Why is ${\Sigma}_{i = 0}^{\infty} \frac{1}{{2}^{i}} = 2$?

Notice that ${a}_{0} = 1$
and each additional term ${a}_{i}$ reduces the distance between the sum up to this point and $2$ by $\frac{1}{2}$