   Chapter 11.4, Problem 15E

Chapter
Section
Textbook Problem

# Determine whether the series converges or diverges.15. ∑ n = 1 ∞ 4 n + 1 3 n − 2

To determine

Whether the series n=14n+13n2 converges or diverges.

Explanation

Given:

The series n=14n+13n2 .

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 geometric series is n=1arn1is convergent if |r|<1 and divergent if |r|>1 .

Calculation:

Consider the given series n=14n+13n2 .

For n1 ,

3n>3n213n<13n2

Thus, 4n+13n<4n+13n2 . (1)

Simplify above equation as follows,

4n+13n=4n43n

Thus, 4n+13n=4(43)n

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