Geometric Series

A series a₁ + a₂ + a₃ + ... is geometric if a_n+1/a_n = r for all n.

Write S_n = a₁ + a₂ + ... + a_n. Then

r*S_n = r*(a₁ + a₂ + ... + a_n) = a₂ + a₃ + ... + a_n+1.

So a₁ + r*S_n = (a₁ + a₂ + ... + a_n) + a_n+1 = S_n + a_n+1

Solving for S_n, we obtain

S_n = (a₁ - a_n+1)/(1 - r)

If |r| < 1, note that lim_{n -> infinity} a_n = 0 and

a₁ + a₂ + a₃ + ... = lim_{n -> infinity}S_n = a₁/(1 - r)

Example 1: 1/5 + 1/10 + 1/20 + 1/40 + 1/80 + 1/160 is a geometric series with a₁ = 1/5, r = 1/2, and n = 6, so the sum is (1/5 - 1/320)/(1 - 1/2) = 63/160.

Example 2: The sum of the infinite geometric series 1/5 + 1/10 + 1/20 + 1/40 + 1/80 + 1/160 + ... is 1/5/(1 - 1/2) = 2/5.

Example 3: 3 + 6 + 12 + 24 + 48 is a geometric series with a₁ = 3, r = 2, and n = 5, so the sum is (3 - 96)/(1 - 2) = 93.