   Chapter 11.1, Problem 56E

Chapter
Section
Textbook Problem

# Determine whether the sequence converges or diverges. If it converges, find the limit..56. a n = ( − 3 ) n n !

To determine

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

Explanation

Given:

The sequence is an=(3)nn! .

Definition used:

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

Theorem used:

If limn|an| converges to zero, then limnan is converges to zero. (1)

Squeeze Theorem:

If xnznyn for nN and limnxn=limnyn=l then limnzn=l . (2)

Calculation:

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

Compute limnan=limn((3)nn!) .

Compute the value of (3)nn! .

|(3)nn!|=|(1)n3nn!|={3nn!,if n is odd3nn!,if n is even=3nn!

Note that, the factorial of n is 123...(n1)n. .

3nn!=333...33n times123...(n1)n=313233..

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Calculus: An Applied Approach (MindTap Course List) 