Question

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

Answer 1

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`