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Sum of a Geometric Series

Infinite sum

sum_(k=0)^∞ x^k = 1/(1 - x) when abs(x)<1

Partial sum formula

sum_(k=0)^n x^k = (x^(n + 1) - 1)/(x - 1)

Series representations

1/(1 - x) = sum_(n=0)^∞ x^n for abs(x)<1

1/(1 - x) = sum_(n=-∞)^∞ ( piecewise | -1 | n = -1
0 | otherwise) (-1 + x)^n