The geometric series is
The convergence of geometric series depends on the value of
- If
, the geometric series diverges - If
, the series converges to
Derivation
Close Form to Summation
See also: Maclaurin series
One way to expand
Then,
These expressions converge iff
Summation to close Form
To express
Let
This algebraic manipulation is valid when
Finite Geometric Series
We get a finite geometric series if we truncating the geometric series into several terms
Related
- 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
: