   Chapter 11.4, Problem 12E

Chapter
Section
Textbook Problem

# Determine whether the series converges or diverges.12. ∑ k = 1 ∞ ( 2 k − 1 ) ( k 2 − 1 ) ( k + 1 ) ( k 2 + 4 ) 2

To determine

Whether the series k=1(2k1)(k21)(k+1)(k2+4)2 converges or diverges.

Explanation

Given:

The series is k=1(2k1)(k21)(k+1)(k2+4)2 .

Result used:

(1) “Suppose that an and bn are the series with positive terms,

(a) If bn is convergent and anbn for all n , then an is also convergent.

(b) If bn is divergent and anbn for all n , then an is also divergent.”

(2) The p-series is convergent if p>1 and diverges if p1 .

Calculation:

Consider the series k=1(2k1)(k21)(k+1)(k2+4)2 .

Then, (2k1)<2k .

(2k1)(k21)<2k(k2)

Thus, (2k1)(k21)(k+1)(k2+4)2<2k(k2)(k+1)(k2+4)2 . (1)

Simplify the terms of the Right-Hand-Side of the inequality as follows,

For k1, 1k+1<1k and 1(k2+4)2<1(k2)2

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