   Chapter 11.3, Problem 26E

Chapter
Section
Textbook Problem

# Determine whether the series is convergent or divergent.26. ∑ n = 1 ∞ n n 4 + 1

To determine

Whether the series is convergent or divergent.

Explanation

Given:

The series is n=1nn4+1 .

Result used:

(1) If the function f(x) is continuous, positive and decreasing on [1,) and let an=f(n) , then the series n=1an is convergent if and only if the improper integral 1f(x)dx is convergent.

(2) The function f(x) is decreasing function if f(x)<0 .

Definition used:

The improper integral abf(x)dx is convergent if the corresponding limit exists.

Quotient Rule: If f1(x) and f2(x) are both differentiable, then

ddx[f1(x)f2(x)]=f2(x)ddx[f1(x)]f1(x)ddx[f2(x)][f2x]2

Calculation:

Consider the function from given series xx4+1 .

The derivative of the function is obtained as follows,

f(x)=(x4+1)ddx(x)xddx(x4+1)(x4+1)2=(x4+1)(1)x(4x3)(x4+1)2=x4+14x4(x4+1)2=13x4(x4+1)2

Since f(x)<0 , the given function is decreasing by using the Result (2).

Clearly, the function f(x) is continuous, positive and decreasing on [1,) .

Use the Result (1), the series is convergent if the improper integral 1xx4+1dx is convergent.

By the definition, the improper integral is convergent if the limit exists.

Compute 1xx4+1dx , as shown below. (1)

Obtain 1t12(2x)1+(x2)2dx .

Consider the indefinite integral 12(2x)1+(x2)2dx

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