   Chapter 11.7, Problem 12E

Chapter
Section
Textbook Problem

Test the series for convergence or divergence. ∑ k = 1 ∞ 1 k k 2 + 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.

And bn is divergent if an is divergent.

2) Given:

k=11kk2+1

3) Calculation:

k+1>k, for all n1

k2+1>k2, for all n1

1k

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