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.



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.