   Chapter 11.1, Problem 45E

Chapter
Section
Textbook Problem

# Determine whether the sequence converges or diverges. If it converges, find the limit.45. an = n sit(1/n)

To determine

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

Explanation

Given:

The sequence is an=nsin(1n) .

Definition used:

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

Limit chain rule:

“If limubf(u)=L and limxag(x)=b with f(x) is said to be continuous at x=b , then the value of limxaf(g(x)) is L.”

Calculation:

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

Compute the value of limnan=limn(nsin(1n)) .

limnan=limn(nsin(1n))         =limn(sin(1n)1n)=sin(1)1=sin(0)0

Since 00 is in indeterminate form, apply L’Hospital’s rule.

limn(nsin(1n))=limn(ddn(sin(1n))ddn(1n))         =limn(ddn(sin(1n))ddn(1n)

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