Chapter 8.5, Problem 26E

a.

To determine

To find: what can be said about the convergence or divergence of the following series.

  ${\displaystyle \sum _{n=0}^{\infty }{c}_n}$.

Answer to Problem 26E

Converges.

Explanation of Solution

Given information: Suppose that ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{x}^n$ converges when x =-4 and diverges when x= 6.

Calculation:

The Centre of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{x}^n$ is 0.

The Interval of convergence is of the form (- k, k )

One or both the end points may be included in the interval of convergence,

Since it is given that the series converges when x =-4, this implies that the series converges for all x which lie in the interval (-4, 4).

The series ${\displaystyle \sum _{n=0}^{\infty }{c}_n}$ is obtained after substituting x = 1 in ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{x}^n$ and x =1 belongs to the interval (-4, 4). Hence the given series converges.

b.

To determine

To find: what can be said about the convergence or divergence of the following series.

  ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{8}^n$.

Answer to Problem 26E

Diverges.

Explanation of Solution

Given information: Suppose that ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{x}^n$ converges when x =-4 and diverges when x= 6.

Calculation:

The Centre of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{x}^n$ is 0.

The Interval of convergence is of the form (- k, k )

One or both the end points may be included in the interval of convergence,

Since it is given that the series diverges when x =6, this implies that the series convergence is smaller than or equal to (-6, 6).

The series ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{8}^n$ is obtained after substituting x = 8 in ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{x}^n$ and x =8 is outside the interval (-6, 6). Hence the given series diverges.

c.

To determine

To find: what can be said about the convergence or divergence of the following series.

  ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{(-3)}^n$.

Answer to Problem 26E

Converges.

Explanation of Solution

Given information: Suppose that ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{x}^n$ converges when x =-4 and diverges when x= 6.

Calculation:

The Centre of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{x}^n$ is 0.

The Interval of convergence is of the form (- k, k )

One or both the end points may be included in the interval of convergence,

Since it is given that the series converges when x =-4, this implies that the series converges for all x which lie in the interval (-4, 4).

The series ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{(-3)}^n$ is obtained after substituting x = -3 in ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{x}^n$ and x =-3 belongs to the interval (-4, 4). Hence the given series converges.

d.

To determine

To find: what can be said about the convergence or divergence of the following series.

  ${\displaystyle \sum _{n=0}^{\infty }{(-1)}^n{c}_n}{9}^n$.

Answer to Problem 26E

Diverges.

Explanation of Solution

Given information: Suppose that ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{x}^n$ converges when x =-4 and diverges when x= 6.

Calculation:

The Centre of convergence of the series ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{x}^n$ is 0.

The Interval of convergence is of the form (- k, k )

One or both the end points may be included in the interval of convergence,

Since it is given that the series diverges when x =6, this implies that the series convergence is smaller than or equal to (-6, 6).

The series ${\displaystyle \sum _{n=0}^{\infty }{(-1)}^n{c}_n}{9}^n={\displaystyle \sum _{n=0}^{\infty }{c}_n}{(-9)}^n$ is obtained after substituting x = -9 in ${\displaystyle \sum _{n=0}^{\infty }{c}_n}{x}^n$ and x =-9 is outside the interval (-6, 6). Hence the given series diverges.