Biboroku

A Trick for Computing the Sum of Geometric Series

Written by Taro Sato, on . Tagged: math

Say if I need to compute the sum of a series like this one:

(1)y=1+2x+3x2+4x3+ ,

where |x|<1. This series looks like a geometric series in which case the sum can be computed from

k=0axk=a1x .

The coefficients vary in Eq. (1), so the relation cannot be directly used.

There is a trick to transform a series like above into a geometric series. Multiplying Eq. (1) by x,

(2)xy=x+2x2+3x3+

Subtracting Eq. (2) from Eq. (1),

y(1x)=1+x+x2+x3+

where the right-hand side is a geometric series with a=1. Hence

y=1(1x)2 .