   Chapter 11.7, Problem 16E

Chapter
Section
Textbook Problem

Test the series for convergence or divergence. ∑ n = 1 ∞ n 4 + 1 n 3 + n

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

Limit comparison test:

an and bn  are the series with positive terms. If limnanbn=c>0,   finite number, then both series are either convergent or divergent.

2) Given:

n=1n4+1n3+n

3) Calculation:

Here, an=n4+1n3+n  is a rational function.

Therefore, use the limit comparison test to test the convergence of given series.

an=n4+1n3+n ,  bn=1n

Consider,

limnanbn=limnn4+

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