   Chapter 11.5, Problem 11E

Chapter
Section
Textbook Problem

Test the series for convergence or divergence. ∑ n = 1 ∞ ( − 1 ) n + 1 n 2 n 3 + 4

To determine

To test:

The given series isconvergent or divergent

Explanation

1) Concept:

The Alternating Series Test:

If the alternating series

n=1-1n+1bn,  bn>0

satisfies,

i) bn+1<bn,  for all n

ii) limnbn=0

then the series is convergent.

2) Given:

n=1-1n+1n2n3+4

3) Calculation:

The given series is in the form ofalternating series.

The given series can be written as

n=1-1n+1n2n3+4=n=1-1n+1bn

where,

bn=n2n3+4

bn>0, for all n1

Now, to check bn is decreasing series or not, by using first derivative test, write bn in the form of function of t.

ft=t2t3+4

Differentiate with respect to t

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