   Chapter 11.1, Problem 77E

Chapter
Section
Textbook Problem

Determine whether the sequence is increasing, decreasing, or not monotonic. Is the sequence bounded? a n = 3 − 2 n e − n

To determine

Whether sequence  an  is increasing, decreasing, or not monotonic and whether it is bounded

Explanation

1) Concept:

Using definitions, identify if the sequence is increasing, decreasing, or not monotonic and find its boundedness.

2) Definition:

i. A sequence  an  is called increasing if an<an+1 for all  n1, that is, a1<a2<a3<

ii. It is called decreasing if an>an+1 for all  n1.

iii. A sequence is monotonic if it is either increasing or decreasing.

iv. Converges: If limnan exists then the sequence is called convergent.

v. Diverges: If limnan does not exist then the sequence is called divergent.

vi. A sequence an is bounded above if there is a number M such that an< M  for all n1.

It is bounded below if there is a number m such that, m <an  for all n1.

If it is bounded above and below, then an  is a bounded sequence.

3) Given:

an=3-2ne-n

4) Calculation:

It is given that

an=3-2ne-n

It can be written in the form of a function:

ft=3-2te-t

Differentiate with respect to t by using the quotient rule:

f't=0--2te-t+2e-t

=2te-t-2e-t

=2e-tt-1

Here f't>0, for all t>1; that means the function is increasing

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