How do you find the sum of an infinite geometric series?

Answer 1

An infinite geometric series can be summed only if the common ratio has an absolute value less than 1.

Suppose the sum to n terms is S(n). That is,

S(n) = a + ar + ar2 + ... + arn-1

Multipying through by the common ratio, r, gives

r*S(n) = ar + ar2 + ar3 + ... + arn

Subtracting the second equation from the first,

S(n) - r*S(n) = a - arn

(1 - r)*S(n) = a*(1 - rn)

Dividing by (1 - r),

S(n) = (1 - rn)/(1 - r)

Now, since |r| < 1, rn tends to 0 as n tends to infinity and so

S(n) tends to 1/(1 - r) or, the infinite sum is 1/(1 - r)