   Chapter 11.1, Problem 43E

Chapter
Section
Textbook Problem

# Determine whether the sequence converges or diverges. If it converges, find the limit.43. a n = cos 2 n 2 n

To determine

Whether the sequence converges or diverges and obtain the limit if the sequence converges.

Explanation

Given:

The sequence is an=cos2n2n .

Definition used:

If an is a sequence and limnan exists, then the sequence an is said to be converges; otherwise it diverges.

Squeeze Theorem:

If xnznyn for nN and limnxn=limnyn=L , then the value of limnzn is L.

Calculation:

Obtain the limit of the sequence to investigate whether the sequence converges or diverges.

Compute the value of limnan=limncos2n2n .

Since 1cosn1 and 0cos2n1 , apply the Squeeze Theorem and obtain the relation as follows:

limn(02n)limn(cos2n2n)limn(12n)

0limn(cos2n2n)limn(12n) (1)

Obtain the value of limn(12n)

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