   Chapter 11.3, Problem 16E

Chapter
Section
Textbook Problem

# Determine whether the series is convergent or divergent.16. ∑ n = 1 ∞ n 1 + n 3 / 2

To determine

Whether the series is convergent or divergent.

Explanation

Given:

The series is n=1n1+n3/2 .

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 n=1x1+x3/2 .

The derivative of the function is obtained as follows,

f(x)=(1+x32)ddx(x)xddx(1+x32)(1+x32)2=(1+x32)(12x12)x12(32x12)(1+x32)2=12x12+12x32x(1+x32)2=12xx(1+x32)2

Simplify further and obtain the derivative.

f(x)=12xx2x(1+x32)2=12xx2x(1+x32)2

Since f(x)<0  for x1 , the given function is decreasing by using the Result (2)

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