power series
A power series has the form:
where both c and a are constants. c must be a nonzero number that is negative or positive, but a may be zero, positive, or negative. x may have a constant inside of the power, such as (2x)^n or (3/4 * x)^n. Note that c is not raised to the power n.
uses and relations to other topics
A power series is much like Taylor/Maclaurin series in that it is used to find the graph of an unusual function. The power series is a type of convergence test as well - if | x | > 1, it will not converge. These functions must have the form:
geometric series
Geometric series are very similar to power series - the only difference is that x (or x - a) is replaced by r, where | r | < 1. Unlike power series (which diverge), geometric series converge. The limit of the geometric series is c / (1 - r). If the sum of the polynomials does not go to infinity, but instead goes to a numerical value, use
to find the sum of the polynomials.
Note: the limit and sum equation only work with geometric series, not power series.
Note: the limit and sum equation only work with geometric series, not power series.
uses and relations to other topics
Geometric series are used to express repeating decimals as fractions!