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

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Answer 1 · 2015-12-22T17:24:27

#-4/3#

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#