Infinite Geometric Series

Infinite Geometric Series

How to solve the infinite geometric sequence problems: formula, proof, examples, and their solutions.

Formula

S = a1/(1 - r). S: infinite sum, a1: first term, r: common ratio

The infinite geometric series means
S = a1 + a2 + a3 + ...
= a1 + a1r + a1r2 + ... .

Geometric series

If -1 < r < 1,
S gets close to a constant number.

S = a1 / (1 - r)

S: infinite sum
a1: first term
r: common ratio

Proof

Infinite Geometric Series: Proof of the Formula

Start from Sn:
Sn = a1(1 - rn) / (1 - r)

Geometric series

If -1 < r < 1 and n → ∞,
then rn becomes 0.

Think of (1/2)n: 1/2, 1/4, 1/8, 1/16, ... .

Then the infinite sum, S, becomes like this:
S = a1(1 - 0) / (1 - r).

So S = a1 / (1 - r).

Example 1

Find the value of the given series. 1 + 1/2 + 1/4 + 1/8 + ...

a1 = 1
r = 1/2(-1 < 1/2 < 1)

S = 1 / (1 - 1/2)

Multiply 2 to both of the numerator and the denominator.

Example 2

Find the value of the given series. The sum of 6*(1/7)^n as n goes from 1 to infinite.

n goes from 1 to ∞.
So use the infinite geometric series formula.

a1 = 6/7
r = 1/7(-1 < 1/7 < 1)

S = (6/7) / (1 - 1/7)