Chapter 11.3, Problem 4E

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

Chapter
Section

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Use the Integral Test to determine whether the series is convergent or divergent.4. ∑ n = 1 ∞ n − 0.3

To determine

Whether the series is convergent or divergent.

Explanation

Given:

The series is n=1an=n=1n0.3 .

Result used: Integral Test

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.

Definition used:

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

Calculation:

Consider the function from given series x0.3 .

The derivative of the function is obtained as follows,

f(x)=(0.3)x0.7=(0.3x0.7)

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

Use the Result (1), the series is divergent if the improper integral 1x0

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