   Chapter 11.3, Problem 20E

Chapter
Section
Textbook Problem

# Determine whether the series is convergent or divergent.20. ∑ n = 3 ∞ 3 n − 4 n 2 − 2 n

To determine

Whether the series is convergent or divergent.

Explanation

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 .

Chain rule: d[f(x)]ndx=n[f(x)]n1f(x)

Given:

The series is n=33n4n22n .

Definition used:

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

Calculation:

Consider the series, n=33n4n22n . (1)

Obtain the partial fraction of 3n4n22n .

3n4n22n=3n4n(n2)

3n4n(n2)=a0(n2)+a1n (2)

Multiply the equation by n(n2) and simplify the terms,

(3n4)n(n2)n(n2)=a0n(n2)(n2)+a1n(n2)n3n4=a0n+a1(n2)3n4=a0n+a1n2a13n4=(a0+a1)n2a1

Equate the coefficient of n and the constant term on both sides,

a0+a1=32a1=4

Solve the above equations and obtain a0=1 and a1=2 .

Substitute 1 for a0 and 2 for a1 in equation (2),

3n4n(n2)=1n2+2n

Therefore, the partial fraction is 3n4n(n2)=1n2+2n (3)

Substitute equation (3) in equation (1), the series is 33n4n(n2)=3(1n2+2n) .

Consider the function from the above series 1x2+2x .

The derivative of the function is obtained as follows:

f(x)=(1)(x2)11ddx(x2)+(1)(x)11ddx(x)=[(x2)2(1)+(x)2(1)]=[(x2)2+(x)2]=[1(x2)2+1x2]

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

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