Question

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

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asked 2021-01-22
Find the sum of the infinite geometric series.
\(1+\frac14+\frac{1}{16}+\frac{1}{64}+...\)

Answers (1)

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 \(\frac43\)

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