Geometric Series

A geometric series is a series for which the ratio of each two consecutive terms is a constant function of the summation index . The more general case of the ratio a rational function of the summation index produces a series called a hypergeometric series.

For the simplest case of the ratio equal to a constant , the terms are of the form . Letting , the geometric sequence with constant is given by

(1)

is given by

(2)

Multiplying both sides by gives

(3)

and subtracting (3) from (2) then gives

			(4)
			(5)

so

(6)

For , the sum converges as ,in which case

(7)

Similarly, if the sums are taken starting at instead of ,

			(8)
			(9)

the latter of which is valid for .