   Chapter 11.3, Problem 19E

Chapter
Section
Textbook Problem

# Determine whether the series is convergent or divergent.19. ∑ n = 1 ∞ n 3 n 4 + 4

To determine

Whether the series is convergent or divergent.

Explanation

Given:

The series is n=1n3n4+4 .

Definition used:

The improper integral abf(x)dx is divergent if the limit does not exist.

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 divergent if and only if the improper integral 1f(x)dx is divergent.

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

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 x3x4+4 .

The derivative of the function is obtained as follows,

f(x)=(x4+4)ddx(x3)x3ddx(x4+4)(x4+4)2=(x4+4)(3x2)x3(4x3)(x4+4)2=12x2x6(x4+4)2=(12x4)x2(x4+4)2

Since f(x)<0    for x>1241.86 , 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 divergent if the improper integral 1x3x4+4dx is divergent.

By the definition, the improper integral is divergent if the limit does not exist.

Compute 1x3x4+4dx as shown below

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