Question

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

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

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 $$=\frac{5}{6}$$