# How do you find the sum of the infinite geometric series 4/5+4/15+4/45...?

Jul 28, 2018

The sum is $= \frac{6}{5}$

#### Explanation:

Let general term of the GP be

${u}_{n} = a {r}^{n}$

The sum of an infinite GP is

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

The sum is

${S}_{\infty} = \frac{a}{1 - r}$ if the common ration is $| r | < 1$

Here,

The first term is $a = \frac{4}{5}$

And

The common ration is $r = \frac{1}{3}$

Therefore,

${S}_{\infty} = \frac{\frac{4}{5}}{1 - \frac{1}{3}} = \frac{4}{5} \cdot \frac{3}{2} = \frac{6}{5}$