How do you find the sum of the infinite geometric series 64,-16,4,-1....?

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Answer 1 · 2016-01-08T10:02:27

#256/5#

Assuming that the series is convergent, let it be #S#. Therefore

#S = 64 - 16 + 4 - 1 + ...#

#= 4^3 - 4^2 + 4^1 - 4^0 + ...#

Then, dividing #S# by 4 gives

#S/4 = 16 - 4 + 1 - 1/4 + ...#

#= 4^2 - 4^1 + 4^0 - 4^{-1} + ...#

Now comes the magic. Observe that

#S = 4^3 - (4^2 - 4^1 + 4^0 - ...)#

#= 4^3 - (S/4)#

#{5S}/4 = 64#

#S = 256/5#