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

Mylo O'Moore 2020-12-15 Answered
Find the sum of the convergent series.
$\sum _{n=0}^{\mathrm{\infty }}\left(-\frac{1}{5}{\right)}^{n}$
You can still ask an expert for help

• Live experts 24/7
• Questions are typically answered in as fast as 30 minutes
• Personalized clear answers

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

liannemdh
Given series :
$\sum _{n=0}^{\mathrm{\infty }}\left(-\frac{1}{5}{\right)}^{n}$
The given series is
$\sum _{n=0}^{\mathrm{\infty }}\left(-\frac{1}{5}{\right)}^{n}=1-\frac{1}{5}+\frac{1}{25}-\frac{1}{125}+....$
The given series is geometric series.
Here, the first term of the series is a=1.
The common difference is given by
$r=\frac{-\frac{1}{5}}{1}$
$r=\frac{-1}{5}$
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
$S=\frac{1}{1-\left(-\frac{1}{5}\right)}$
$S=\frac{1}{1+\frac{1}{5}}$
$S=\frac{5}{6}$
Therefore, the sum of series is $=\frac{5}{6}$
Jeffrey Jordon