# How do you find the sum of the infinite geometric series 0.5 + 0.05 + 0.005 + ...?

##### 2 Answers

Feb 21, 2016

We notice that

But

first term

hence its sum is given by

So the initial sum is

Feb 21, 2016

#### Explanation:

The sum to n terms of a geometric sequence is

#S_n =( a( 1 - r^n ))/(1 - r) # As

# n → ∞ " then " r^n → 0 # and

# S_∞ = a/(1 - r )# [ for -1 < r < 1 ]where a , is the first term and r , the common ratio

here a =

#0.5 = 1/2 " and " r = 0.5/0.5 = 0.005/0.05 = 1/10 #

#rArr S_∞ =( 1/2)/(1 - 1/10) =( 1/2)/(9/10) = 1/2xx10/9 = 5/9 #