Chapter 9.5, Problem 50E

a.

To determine

To find: To find the interval of convergence.

Answer to Problem 50E

The interval of convergence is $(\frac{1}{e},e)$ .

Explanation of Solution

Given:

The given series ${\displaystyle \sum _{n=N}^{\infty }{(\ln x)}^n}$ .

Formula used:

Sum is given for an infinite geometric progression is given by $S=\frac{a}{1-r}$ where a is first and r is common ratio, also $-1<r<1$ .

Calculation:

Given series is ${\displaystyle \sum _{n=N}^{\infty }{(\ln x)}^n}$

This is an infinite geometric progression.

The series converges absolutely for $\left|\ln x\right|<1$ or $e^{-1}<x<e$ and diverges for $\left|\ln x\right|\ge 1$

The interval of convergence is $(\frac{1}{e},e)$ .

b.

To determine

To find: To find the interval of convergence

Answer to Problem 50E

The interval of convergence is $(\frac{1}{e},e)$ .

Explanation of Solution

Given:

The given series ${\displaystyle \sum _{n=N}^{\infty }{\left|(\ln x)\right|}^n}$ .

Formula used:

Sum is given for an infinite geometric progression is given by $S=\frac{a}{1-r}$ where a is first and r is common ratio, also $-1<r<1$ .

Calculation:

Given series is ${\displaystyle \sum _{n=N}^{\infty }{\left|(\ln x)\right|}^n}$ .

This is an infinite geometric progression.

The series converges absolutely for $\left|\ln x\right|<1$ or $e^{-1}<x<e$ and diverges for $\left|\ln x\right|\ge 1$

The interval of convergence is $(\frac{1}{e},e)$ .

c.

To determine

To find: To find whether the given series converges or diverges.

Answer to Problem 50E

The series converges.

Explanation of Solution

Given:

The given series ${\displaystyle \sum _{n=N}^{\infty }{\left|(\ln x)\right|}^n}$

Formula used:

  $a=f(n)$where $f$ is a continuous, positive, decreasing function of x for all $x\ge N$ (N is a positive integer).

Then the series ${\displaystyle \sum _{n=N}^{\infty }{a}_n}$ and the integral ${\displaystyle {\int }_N^{\infty }f(x)dx}$ either both converges or both diverges.

Calculation:

Given series is ${\displaystyle \sum _{n=N}^{\infty }{\left|(\ln x)\right|}^n}$ .

From above two part, we can observe that the at all points of interval of convergence, series converges absolutely.