Question

# Find the sum of the infinite geometric series. 1+frac14+frac{1}{16}+frac{1}{64}+...

Series
Find the sum of the infinite geometric series.
$$1+\frac14+\frac{1}{16}+\frac{1}{64}+...$$

2021-01-23

To find:
The sum of infinite geometric series.
Given:
The geometric series is $$1+\frac14+\frac{1}{16}+\frac{1}{64}+...$$
Concept used:
The sum of infinite term of the geometric series is
$$S_\infty=\frac{a}{1-r}\quad(r<1)$$
Here, a is first term, r is common ratio( less than 1) and $$S_\infty$$ is the sum of the infinite series.
Calculation:
The first term of the geometric series is 1.
The common ratio can be obtained by the ratio of second term by first term.
$$\frac{\frac14}{1}=\frac14$$
The common ratio $$\frac14$$ is less than 1.
Substitute $$\frac{1}{4}$$ for r and 1 for a in equation
$$S_{\infty}=\frac{1}{1-\frac14}$$
$$=\frac{1}{\frac{4-1}{4}}$$
$$=\frac{4}{3}$$
Thus, the sum of infinite geometric series is $$\frac43$$