Math 242 Enrichment Problems Section 9.1
Monotonic Sequences
In calculus, an important characteristic of a sequence is whether it converges or not. To define this term
requires an understanding of limits, which is beyond the scope of this class. However, a number of
theorems pertaining to the convergence of a sequence depend upon how the
sequence behaves. For example, based on the behavior of the sequence of radii of
the inscribed circles shown in the diagram, we know that the circles will only get
so small.
When describing the behavior of sequences, we use terminology similar to
the language introduced in Chapter 1. The definitions are slightly different due to
the discrete domains of sequences. The symbol ` represents the set of natural
numbers.
DEFINITION
Let {an} be a sequence defined on `, and let N∈` be any natural number.
(1) If an+1 > an for all n ≥ N, then we say that {an} is increasing for n ≥ N.
(2) If an+1 < an for all n ≥ N, then we say that {an} is decreasing for n ≥ N.
In addition, if an+1 ≥ an for all n ≥ N, we say the sequence is monotonically increasing for n ≥ N.
Conversely, if an+1 ≤ an for all n ≥ N, we say the sequence is monotonically decreasing for n ≥ N.
Sometimes the term strictly monotonic is used with sequences that are increasing or decreasing (the
word strictly indicates that consecutive terms are not allowed to be equal).
EXAMPLE 1 Identifying Monotonic Sequences
Determine whether the sequence
31
!
n
n
bn
+
= appears to be monotonic by examining the behavior of the
first eight terms numerically and graphically.
SOLUTION
Looking at the table and graph of the first eight terms, at first the sequence is increasing. However, this
trend only lasts for the first three terms, after which we can see the terms begin to get smaller.
n bn
1 2 2.000
2 9/2 4.500
3 14/3 4.667
4 65/24 2.708
5 21/20 1.050
6 217/720 0.301
7 43/630 0.068
8 57/4480 0.013
Based on this observation, the sequence appears to be decreasing (a strictly monotonic decreasing
sequence) for n ≥ 3.
n
bn
