power series

A power series is a series in the following form

where

• is a variable
• is a constant, often called the center of the series
• are constant coefficients

Convergence

The convergence of a power series depends on the variable , and a power series may converge for some but diverge for others.

There exists a non-negative number , called the radius of convergence, such that:

1. The series converges absolutely for all satisfying
2. The series diverges for all satisfying

The behavior of the series when depends on the specific series.

The set of all for which the series converges is called the interval of convergence. It’s typically an open interval , but may include one or both endpoints.

See Also

• A geometric series is a special case of power series with and
• The Borel’s Theorem states that every power series is the Taylor series of some smooth function