   Chapter 11.3, Problem 20E

Chapter
Section
Textbook Problem

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

To determine

Whether the series is convergent or divergent

Explanation

1) Concept:

i) Integral test:

Suppose f is a continuous, positive, decreasing function on [1, ) and let an=fn. Then series n=1an is convergent if and only if the improper integral 1f(x)dx is convergent.

a) 1fxdx  is convergent, then n=1an is convergent

b) 1fxdx  is divergent, then n=1an is divergent

ii) Improper integral of infinite intervals:

If atfxdx exists for every number ta, then

af(x)dx=limtatfxdx

provided this limit exists (as a finite number)

2) Given:

n=33n-4n2-2n

3) Calculation:

According to concept,

an=fn=3n-4n2-2n

fx=3x-4x2-2x

For, intervals [3, ), the function is positive and continuous

To determine the given function for decreasing, differentiate f(x) with respect to x

f'x=-3x2-8x+8x2-2x2<0

Therefore, function is decreasing so the integral test applies.

33x-4x2-2xdx =limt3t3x-4x2-2xdx

By using partial fraction,

3x-4x(x-2)=Ax+Bx-2

By solving it give, A=2, B=1

3x-4x(x-2

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