5.3.2. Power series

Functional range of the form:

called a power series. The constant numbers a 0 , a 1 , a 2 , ..., a n ... are called the coefficients of the series. The constant number x 0 is called the center of the series.

The region of convergence of a power series is the interval , which is called the convergence interval of a power series, the number R is called the radius of convergence. On the convergence interval, the power series converges absolutely, with power series diverges.

When x = R , x = -R, the question of the convergence of a power series is solved individually for each specific series. Within the interval of convergence, power series can be differentiated and integrated term by term.

The radius of convergence is determined by the following limits: