Question

Find the sum of the convergent series. sum_{n=0}^infty(-frac15)^n

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asked 2020-12-15
Find the sum of the convergent series.
\(\sum_{n=0}^\infty(-\frac15)^n\)

Answers (1)

2020-12-16
Given series :
\(\sum_{n=0}^\infty(-\frac15)^n\)
The given series is
\(\sum_{n=0}^\infty(-\frac15)^n=1-\frac{1}{5}+\frac{1}{25}-\frac{1}{125}+....\)
The given series is geometric series.
Here, the first term of the series is a=1.
The common difference is given by
\(r=\frac{-\frac15}{1}\)
\(r=\frac{-1}{5}\)
The formula to find the sum of infinite geometric series is given by
\(S=\frac{a}{1-r}\)
Substitute the value of "a" and "r" in the above formula, we get
\(S=\frac{1}{1-(-\frac15)}\)
\(S=\frac{1}{1+\frac15}\)
\(S=\frac{5}{6}\)
Therefore, the sum of series is \(=\frac{5}{6}\)
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