The geometric series is

The convergence of geometric series depends on the value of .

  • If , the geometric series diverges
  • If , the series converges to

The geometric series of 1/2 + 1/4 + ...

Derivation

Close Form to Summation

See also: Maclaurin series One way to expand to a series is to use Taylor expansion. Let . Then

Then,

These expressions converge iff tends toward zero, i.e. .

Summation to close Form

To express in close form, we can do some substitution as typical for power series.

Let , then , and

This algebraic manipulation is valid when .

Finite Geometric Series

We get a finite geometric series if we truncating the geometric series into several terms For , it has the close form of

  • We can solve the infamous 0.999… = 1 question by treating as a geometric series.
  • The ratio test can be used to prove the convergence for :