How do you find the sum of the infinite geometric series 3-1+1/3-1/9+...?

1 Answer
Nov 29, 2015

Apply the geometric sum formula to find that
#sum_(n=0)^oo3(-1/3)^n = 9/4#

Explanation:

Given a geometric series with initial value #a# and common ratio #r# with #|r| < 1# the sum is given by the formula

#sum_(n=0)^ooar^n = a/(1-r)#

(A short derivation of this formula is included in Can an infinite series have a sum? )

In the given series, the initial value is #a=3# and the common ratio, found by dividing any term by its preceding term, is #r = -1/3#.

As #|-1/3| < 1# we can apply the formula to obtain

#sum_(n=0)^oo3(-1/3)^n = 3/(1-(-1/3)) = 3/(4/3) = 9/4#