A

David

Darling

power series

A power series is a series of the form

 

general form of a power series

 

The numbers a1, a2, a3, ... are called the coefficients of the power series. x is a variable.

 

Polynomials of the form

 

    a0 + a1x + a2x2 + ... + apxp

 

are also power series with finite numbers of terms.

 


Examples

The binomial series, the exponential series, the logarithmic series, the trigonometric series, and the inverse trigonometric series are power series in the variable x.

 

Every power series possesses an interval of convergence and a radius of convergence. If r is the radius of convergence of a power series, the series converges for all values of x for which |x| < r, while the series diverges for all values of |x| > r. The interval -r < x < +r is referred to as the interval of convergence. The radius of convergence can be 0, in which case the series only converges for the value x = 0. The radius of convergence may also be ∞, in which case the series converges for all values of x.

 

The exponential series

 

exponential series

 

converges for all values of x.

 

The logarithmic series

 

logarithmic series

 

converges for -1 < x ≤ 1.

 

The power series

 

power series

 

converges only for for x = 1.