Chapter 8, Problem 5RCC

a.

To determine

To describe the test for Divergence.

Answer to Problem 5RCC

If the limit of $a_n$ when $n$ approaches to infinity exist or does not exist, but it must be a non-zero number then the infinite series ${\displaystyle \sum _n{a}_n}$ is said to be diverging.

Explanation of Solution

Given information:

test for Divergence.

If the limit of $a_n$ when $n$ approaches to infinity exist or does not exist, but it must be a non-zero number then the infinite series ${\displaystyle \sum _n{a}_n}$ is said to be diverging.

Therefore,

  $\underset{n\to \infty }{\lim }{a}_n\ne 0$ then the series diverges.

b.

To determine

To describe the integral test.

Answer to Problem 5RCC

Integral test is defined as the test in which a non-negative function which is defined on the unbounded interval $\left[N,\infty \right)$ on which it is monotone decreasing, is said to be converging if and only if the improper integral ${\displaystyle \underset{N}{\overset{\infty }{\int }}f\left(x\right)dx}$ is finite.

Explanation of Solution

Given information:

To describe the integral test.

Let, $N$ be an integer and non-negative function $f$ defined on the unbounded interval $\left[N,\infty \right)$ on which it is monotone decreasing, the infinite series is defined by,

  ${\displaystyle \sum _{n=N}^{\infty }f\left(n\right)}$

And it converges to a real number if and only if the improper integral ${\displaystyle \underset{N}{\overset{\infty }{\int }}f\left(x\right)dx}$ is finite.

Hence,

Integral test is defined as the test in which a non-negative function which is defined on the unbounded interval $\left[N,\infty \right)$ on which it is monotone decreasing, is said to be converging if and only if the improper integral ${\displaystyle \underset{N}{\overset{\infty }{\int }}f\left(x\right)dx}$ is finite.

c.

To determine

To describe the comparison test.

Answer to Problem 5RCC

Comparison test is defined as the test in which one series or integral whose convergence properties is to be determined is compared with the series or integral whose convergence properties is known.

Explanation of Solution

Given information:

the comparison test.

Comparison test is defined as the test in which one series or integral whose convergence properties is to be determined is compared with the series or integral whose convergence properties is known.

For, series, the comparison test is defined by,

If the infinite series ${\displaystyle \sum _{}{b}_n}$ converges and $0\le a_n\le b_n$ for all larger value of $n$ , then the infinite series ${\displaystyle \sum _{}{a}_n}$ is also said to be convergent series. Similarly, If the infinite series ${\displaystyle \sum _{}{b}_n}$ diverges and $0\le b_n\le a_n$ for all larger value of $n$ , then the infinite series ${\displaystyle \sum _{}{a}_n}$ is also said to be divergent series

d.

To determine

To describe the limit comparison test.

Answer to Problem 5RCC

Limit comparison test is defined as the test in which the two series ( ${\displaystyle \sum _n{a}_n}$ and ${\displaystyle \sum _n{b}_n}$ where $a_n\ge 0\text{ and }{b}_n\ge 0$ for all $n$ ) is said to be either divergent series or convergent series if $\underset{n\to \infty }{\lim }\frac{a_n}{b_n}=c$ with $0<c<\infty$ . It is used for testing the convergence of an infinite series

Explanation of Solution

Given information:

The limit comparison test.

Limit comparison test is defined as the test in which the two series ( ${\displaystyle \sum _n{a}_n}$ and ${\displaystyle \sum _n{b}_n}$ where $a_n\ge 0\text{ and }{b}_n\ge 0$ for all $n$ ) is said to be either divergent series or convergent series if $\underset{n\to \infty }{\lim }\frac{a_n}{b_n}=c$ with $0<c<\infty$ . It is used for testing the convergence of an infinite series

e.

To determine

To describe the alternating series test.

Answer to Problem 5RCC

Alternating series test is defined as the test in which an alternating series $a_n$ is said to be convergent series if and only if $\left|a_n\right|$ decreases monotonically and $\underset{n\to \infty }{\lim }{a}_n=0$ .

Explanation of Solution

Given information:

The alternating series test.

Consider, an alternating series,

  ${\displaystyle \sum _{n=0}^{\infty }{\left(-1\right)}^n}{a}_n=a_0-a_1+a_2-a_3+\mathrm{......}$

Where, $a_n$ is either all positive or all negative.

Alternating series test is defined as the test in which an alternating series $a_n$ is said to be convergent series if and only if $\left|a_n\right|$ decreases monotonically and $\underset{n\to \infty }{\lim }{a}_n=0$ .

f.

To determine

To describe the ratio test.

Answer to Problem 5RCC

The ratio test use the limit of form $L=\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}}{a_n}\right|$  

The ratiotest states that if  $L<1$ then the series converges absolutely; if  $L>1$ then the series is divergent; if  $L=1$ or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.

Explanation of Solution

Given information:

the ratio test.

The ratio test use the limit of form $L=\underset{n\to \infty }{\lim }\left|\frac{a_{n+1}}{a_n}\right|$  

The ratiotest states that if  $L<1$ then the series converges absolutely; if  $L>1$ then the series is divergent; if  $L=1$ or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.