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

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

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

${5S}/4 = 64$

$S = 256/5$