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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.

Answer and Explanation:


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}.


Learn more about this topic:

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Understand the Formula for Infinite Geometric Series

from Algebra II Textbook

Chapter 21 / Lesson 11
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