# How do you find the sum of the infinite series Sigma7(1/10)^k from k=1 to oo?

Feb 18, 2017

$\frac{7}{9}$

#### Explanation:

This is equal to $7 {\sum}_{1}^{\infty} \frac{1}{10} ^ k$

This is now a geometric series with first term $\frac{1}{10}$ and common ratio $\frac{1}{10}$ or say, both are 0.1

Therefore, the sum of this series would be $7 \frac{0.1}{1 - 0.1}$
[formula for sum of an infinite geometric series is $\frac{a}{1 - r} , w h e r e , r < 1$)

$\frac{7 \left(0.1\right)}{0.9} = \frac{7}{9}$