Question

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

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

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