# Find the sum of the infinite geometric series. 4 + 1 + \frac{1}{4}+....

## Question:

Find the sum of the infinite geometric series.

{eq}4 + 1 + \frac{1}{4}+.... {/eq}

## Geometric Series

The geometric series constitute two terms, which defines the series, first is a common ratio, and the second is the first term. Any number of a geometric series is calculated by multiplying the common ratio with the previous number.

Given Data

• The given geometric series is: {eq}4 + 1 + \dfrac{1}{4} + ....... {/eq}

The general expression for the infinite geometric series is,

{eq}GP = a + ar + a{r^2} + a{r^3} + ........ {/eq}

Here, the common ratio is {eq}r {/eq} and the first term is {eq}a {/eq}.

Compare the give series with the standard series,

{eq}\begin{align*} a &= 4\\ r &= \dfrac{1}{4} \end{align*} {/eq}

The expression for the sum of the infinite geometric series is,

{eq}S = \dfrac{a}{{1 - r}} {/eq}

Substitute the known values,

{eq}\begin{align*} S &= \dfrac{4}{{1 - \left( {\dfrac{1}{4}} \right)}}\\ &= \dfrac{4}{{\left( {\dfrac{3}{4}} \right)}}\\ &= \dfrac{{16}}{3} \end{align*} {/eq}

Thus, the sum of the infinite geometric series is {eq}\dfrac{{16}}{3} {/eq}. 