Question

(a) Define the radius of convergence of power series and how to find it.

(b) Define the interval of convergence of a power series.

Short answer

(a) The definition is stated below.

(b) The definition is stated below

Step-by-step solution

Definition of radius of convergence of power series. (a)

The radius of convergence of a power series is a positive number\(R\).

Three possibilities of power series.(a)

For the power series\(\sum\limits_{n = 0} {{c_n}} {(x - a)^n}\), there are three possibilities:

1. \(0\)if the power series converges only when \(x = a\).
2. \(\infty \)if the series converges for all \(x\), or
3. A positive number \(R\)such that the series converges if \({\rm{|x - a| < R}}\)and diverges if \({\rm{|x - a| > R}}\).

The radius of the convergence of the power series \(R\) can be obtained by using the ratio test.

Definition of the interval of convergence of power series.(b) 

The interval of convergence of a power series is the interval that consists of all values of\({\rm{x}}\)in which the series converges.

: Interval of convergence with respect to part (a).(b)

The interval of convergence with respect to the cases in part (a) is,

1. The single point\((a)\).
2. All real numbers, that is the real numbers line\( - \infty < x < \infty \).
3. An interval with end points \(a - R\) and \(a + R\)which can contain neither, either, or both the end points.

That is, there are four possibilities for the interval of convergence:

\((a - R,a + R),(a - R,a + R),\;\;\;(a - R,a + R)\) and\({\rm{(a - R, a + R)}}\)