   Chapter 11.6, Problem 6E

Chapter
Section
Textbook Problem

# Determine whether the series is absolutely convergent or conditionally convergent.6. ∑ n = 1 ∞ ( − 1 ) n − 1 n n 2 + 4

To determine

Whether the series is absolutely convergent or conditionally convergent.

Explanation

Definition used:

“A series an is called absolutely convergent if the series of absolute values |an| is convergent.”

“A series an is called conditionally convergent if it is convergent but not absolutely convergent.”

Result used:

(1) “If the alternating series n=1(1)n1bn=b1b2+b3b4+...   bn>0 satisfies the conditions, bn+1bn for all n and limnbn=0, then the series is convergent.”

(2) “Suppose that an and bn are the series with positive terms, if limnanbn=c, where c is a finite number and c>0, then either both series converge or both diverge.”

(3) The function f(x) is decreasing when f(x)<0.

Calculation:

Consider the given series n=1(1)n1nn2+4 where an=nn2+4>0 for n1.

Obtain the derivative of the function f(x)=xx2+4.

f(x)=ddx(xx2+4)=(x2+4)ddx(x)xddx(x2+4)(x2+4)2=x2+4(1)x(2x)(x2+4)2=x2+4(x2+4)2

Since f(x)<0 for x2 and by the Result (3), the function is a decreasing function.

That is, an is decreasing when n2. (1)

Obtain the limit of an.

limnan=limnnn2+4=limnnn(n+4n)=limn1(n+4n)=0    [1=0]

That is, limnan=0

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