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

1 Answer
Dec 22, 2015

-4/3

Explanation:

The formula for the sum of a geometric series is

S = r/(1 - r)

where r is the ratio between the successive terms of the series.

Let's look at the terms to find out what the ratio is. Divide 1 by -2. (the second term by the first term)

The ratio is -1/2.

Indeed, divide -1/2 (3rd term) by 1 (2nd term) and you'll also get -1/2.

So, let's plug our ratio into the formula:

S = (-1/2)/(1 - (-1/2))
S = -1/3

Be careful here - this is the sum of a series whose first term is the ratio itself. In other words, this is the sum of the series

-1/2 + 1/4 - 1/8 + 1/16 - ... = S

To get the correct sum, all we need to do is add -2 + 1 to S.

-1/3 - 2 + 1 = -4/3