Chapter 8, Problem 2RCC

a.

To determine

To describe what is a bounded sequence.

Answer to Problem 2RCC

A sequence is bounded if it is bounded above and below, that is to say, if there is a number, k, less than or equal to all the terms of sequence and another number, K', greater than or equal to all the terms of the sequence. Therefore, all the terms in the sequence are between k and K'.

Explanation of Solution

Given information:

Bounded Sequence: A sequence is bounded if it is bounded above and below, that is to say, if there is a number, k , less than or equal to all the terms of sequence and another number, K' , greater than or equal to all the terms of the sequence.

Therefore, all the terms in the sequence are between k and K' .

b.

To determine

To describe what is a monotonic sequence.

Answer to Problem 2RCC

The monotonic sequence are the sequence which are eitheran increasing sequence or a decreasing sequence. A sequence $a_n$ is considered as an increasing sequence if $a_n<a_{n+1}$ for every $n$ and A sequence $a_n$ is considered as an decreasing sequence if $a_n>a_{n+1}$ for every $n$

Explanation of Solution

Given information:

A monotonic sequence.

The monotonic sequence are the sequence which are eitheran increasing sequence or a decreasing sequence. A sequence $a_n$ is considered as an increasing sequence if $a_n<a_{n+1}$ for every $n$ and A sequence $a_n$ is considered as an decreasing sequence if $a_n>a_{n+1}$ for every $n$

c.

To determine

To describe a bounded monotonic sequence.

Answer to Problem 2RCC

If there exists a number $m$ such that $m\le a_n$ for every $n$ then the monotonic sequence is bounded below and if there exists a number $M$ such that $a_n\le M$ for every $n$ then the monotonic sequence is bounded above.

Explanation of Solution

Given information:

A bounded monotonic sequence.

If there exists a number $m$ such that $m\le a_n$ for every $n$ then the monotonic sequence is bounded below and if there exists a number $M$ such that $a_n\le M$ for every $n$ then the monotonic sequence is bounded above.