   Chapter 11.1, Problem 61E

Chapter
Section
Textbook Problem

Use a graph of the sequence to decide whether the sequence is convergent or divergent. If the sequence is convergent, guess the value of the limit from the graph and then prove your guess. (See the margin note on page 739 for advice on graphing sequences.) a n = n 2  cos  n 1 + n 2

To determine

To decide:

a) The sequence converges or diverges

b) If it converges, find the limit

Explanation

1) Concept:

By observing the graph, identify whether the sequence is convergent or divergent and also identify the limit if the sequence is convergent

2) Definition:

Convergent: If limnan exists then the sequence is called convergent

Divergent: If limnan does not exist then the sequence is called divergent

3) Given:

an=n2cosn1+n2

4) Calculation:

Consider the given sequence,

an=n2cosn1+n2

Draw the graph of the given sequence

From the graph,

The given sequence is divergent because the terms of the sequence oscillate between -1 & 1  approximately

Now take limit on the given sequence as  n,

limnan=limnn2cos&

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