   Chapter 11.1, Problem 49E

Chapter
Section
Textbook Problem

# Determine whether the sequence converges or diverges. If it converges, find the limit.49. an = ln(2n2 + 1) − ln(n2 + 1)

To determine

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

Explanation

Given:

The sequence is an=ln(2n2+1)ln(n2+1) .

Definition used:

If an is sequence and limnan exists, then the sequence an is said to be converges otherwise it is 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 limn(ln(2n2+1)ln(n2+1)) .

Apply the Logarithm Quotient rule, logc(a)logc(b)=logc(ab) .

limn(ln(2n2+1)ln(n2+1))=limn(ln(2n2+1n2+1))

Consider the functions g(n)=2n2+1n2+1 and f(u)=ln(u) and obtain limng(n) .

limng(n)=limn(2n2+1n2+1)

Divide numerator and the denominator by the highest power.

limn(2n2+1n2+1)=limn(2n2+1n2n2+1n2)             =limn(2+1n21+1n2)

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