   Chapter 11.3, Problem 8E

Chapter
Section
Textbook Problem

# Use the Integral Test to determine whether the series is convergent or divergent.8. ∑ n = 1 ∞ n 2 e − n 3

To determine

Whether the series is convergent or divergent.

Explanation

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

Product Rule: ddx[f1(x)f2(x)]=f1(x)ddx[f2(x)]+f2(x)ddx[f1(x)]

Given:

The series is n=1an=n=1n2en2 .

Definition used:

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

Calculation:

Consider the function from the given series x2ex2 .

The derivative of the function is computed as follows:

f(x)=x2ddx(ex2)+ex2ddx(x2)=x2ex2ddx(x2)+ex2(2x)=x2ex2(

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