   Chapter 11.3, Problem 7E

Chapter
Section
Textbook Problem

# Use the Integral Test to determine whether the series is convergent or divergent.7. ∑ n = 1 ∞ n n 2 + 1

To determine

Whether the series is convergent or divergent.

Explanation

Result used:

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.

Derivative rule: 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)][f2(x)]2

Given:

The series is n=1an=n=1nn2+1 .

Definition used:

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

Calculation:

Consider the function from given series xx2+1 .

The derivative of the function is obtained as follows,

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

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