   Chapter 11.7, Problem 20E

Chapter
Section
Textbook Problem

Test the series for convergence or divergence. ∑ k = 1 ∞ k 3 − 1 k ( k + 1 )

To determine

To test:

The series for convergence or divergence.

Explanation

1) Concept:

i) If the series has a form that is similar to a p-series or a geometric series, then one of the comparison tests should be considered. In particular, if an is a rational function or an algebraic function of n(involving roots of polynomials), then the series should be compared with a p-series.

ii) A p-series 1/np converges if p>1 and diverges if p1

Comparison Test:

an and bn are series with positive terms

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

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

2) Given:

k=1k3-1kk+1

3) Calculation:

Here,

ak=k3-1kk+1 is rational function of k with positive terms

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