   Chapter 11.1, Problem 35E

Chapter
Section
Textbook Problem

# Determine whether the sequence converges or diverges. If it converges, find the limit.35. a n = ( − 1 ) 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=(1)n2n .

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|=0 , then limnan=0 . (1)

Results used:

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

(2) If f(x) is a function and c is any constant, then, limxa[cf(x)]=climxaf(x) .

(3) If f(x) is a functions then, limxa[f(x)]b=[limxaf(x)]b .

Calculation:

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

Compute limnan=limn(1)n2n . (2)

The value of (1)n2n is,

|(1)n2n|=|(1)n||2n|={|1||2n|if n is even|1||2n|

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