   Chapter 11.1, Problem 47E

Chapter
Section
Textbook Problem

# Determine whether the sequence converges or diverges. If it converges, find the limit.47. a n = ( 1 + 2 n ) n

To determine

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

Explanation

Given:

The sequence is an=(1+2n)n .

Definition used:

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

Laws of limits for sequences used:

If f(x) and g(x) are two functions, then limxa[f(x)g(x)]=limxaf(x)limxag(x) and limxag(x)0 .

Calculation:

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

Compute the value of limnan=limn(1+2n)n .

Apply the exponent rule ax=exln(a) and simplify the expressions.

limn((1+2n)n)=limn(enln(1+2n))

=elimnnln(1+2n) (1)

Obtain the value of limnnln(1+2n) .

limnnln(1+2n)=limn(11nln(1+2n))=limn(ln(1+2n)1n)=ln(1+2)1=ln(1+0)0

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

limnnln(1+2n)=limn(ddn(ln(1+2n))ddn(1n))                         =limn(ddn(ln(1+2n))

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