How do you find the sum of the infinite geometric series 8 - 4 + 2 - 1 + 1/2 -...?

1 Answer
May 27, 2016

#8-4+2-1+1/2-... =16/3#

Explanation:

A geometric series is a series of the form #sum_(k=0)^ooar^n# where #a# is the first term in the sum and #r# is the constant ratio between terms. The series will converge if and only if #|r|<1# (with the trivial exception being if #a=0#.

Given a convergent geometric series, that is, a geometric series with #|r|<1#, we have

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

(see the above link for a derivation of this formula)

In the given sum, the common ratio between terms is #-1/2#. As #|-1/2|=1/2<1#, the series will converge and we can find the value it converges to with the above formula. Noting that the first term is #a=8#, we have

#8-4+2-1+1/2-... = sum_(n=0)^oo8(-1/2)^n#

#=8/(1-(-1/2))#

#=8/(3/2)#

#=16/3#