Question

Write out the first few terms of the series sum_{n=0}^inftyfrac{(-1)^n}{5^n} What is the​ series' sum?

Series
ANSWERED
asked 2021-02-21
Write out the first few terms of the series
\(\sum_{n=0}^\infty\frac{(-1)^n}{5^n}\)
What is the​ series' sum?

Answers (1)

2021-02-22
Consider the series
\(\sum_{n=0}^\infty\frac{(-1)^n}{5^n}\)
\(\sum_{n=0}^\infty\frac{(-1)^n}{5^n}=\frac{(-1)^0}{5^0}+\frac{(-1)^1}{5^1}+\frac{(-1)^2}{5^2}+\frac{(-1)^3}{5^3}+...\)
\(\sum_{n=0}^\infty\frac{(-1)^n}{5^n}=\frac{1}{1}+\frac{-1}{5}+\frac{1}{5^2}+\frac{-1}{5^3}+...\)
\(\sum_{n=0}^\infty\frac{(-1)^n}{5^n}=1-\frac{1}{5}+\frac{1}{5^2}-\frac{1}{5^3}+...\)
Series sum
\(\sum_{n=0}^\infty\frac{(-1)^n}{5^n}=1-\frac{1}{5}+\frac{1}{5^2}-\frac{1}{5^3}+...\)
It is a infinite geometric series with first term \(a=1\) and common ratio \(r=-\frac{1}{5}\)
Sum of a infinite geometric series is given by
\(S=\frac{a}{1-r}\)
\(S=\frac{1}{1-(-\frac{1}{5})}\)
\(S=\frac{1}{1+\frac{1}{5}}\)
\(S=\frac{1}{\frac{5+1}{5}}\)
\(S=\frac{5}{6}\)
First term is \(=1\)
Sum \(=\frac{5}{6}\)
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