   Chapter 11.7, Problem 2E

Chapter
Section
Textbook Problem

Test the series for convergence or divergence. ∑ n = 1 ∞ n − 1 n 3 + 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. p- series 1np  is convergent if, p>1 and divergent if, p1

Comparison test:

If an  bn  for all n and bn is convergent then, an is also convergent.

2) Given:

n=1n-1n3+1

3) Calculation:

Here, an=n-1n3+1 is a rational function.

Therefore, compare the series with a p-series

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