Question

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

Answer 1

`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`