Chapter 8, Problem 8RCC

a.

To determine

To state the general form of power series.

Answer to Problem 8RCC

The general form of power series is:

  ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{(x-a)}^n=c_0+c_1(x-a)+c_2{(x-a)}^2+c_3{(x-a)}^3+\mathrm{..........}$

Explanation of Solution

Given information:

The general form of power series.

Consider, the series,

  ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{(x-a)}^n=c_0+c_1(x-a)+c_2{(x-a)}^2+c_3{(x-a)}^3+\mathrm{..........}$

Where x is a variable and the $c_n$ ’s are constants called the coefficients of the series anda is a number. For each fixed x , the series is a series of constants that we can test for convergence or divergence.

The general form of power series is:

  ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{(x-a)}^n=c_0+c_1(x-a)+c_2{(x-a)}^2+c_3{(x-a)}^3+\mathrm{..........}$

b.

To determine

To define the radius of convergence of power a series.

Answer to Problem 8RCC

Radius of convergence of a power series may be defined as the radius of largest circle in which the power series converges. It may be a real number or $\infty$ .

Explanation of Solution

Given information:

the radius of convergence of power a series, to write.

Use definition of radius of convergence of a power series.

Radius of convergence of a power series may be defined as the radius of largest circle in which the power series converges. It may be a real number or $\infty$ .

c.

To determine

To define the interval of convergence of a power series.

Answer to Problem 8RCC

Interval of convergence of power series: At the expansion point there is a set of point where the power series converges and this set of points is known as interval of convergence of power series.

Explanation of Solution

Given information:

the interval of convergence of a power series,

At the expansion point there is a set of point where the power series converges and this set of points is known as interval of convergence of power series.