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

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

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