1549901369-Elements_of_Real_Analysis__Denlinger_

90 Chapter 2 • Sequences

(e) monotone if it is any one of (a) or (b) or (c) or (d).

(f) strictly monotone if it is either (c) or (d).

Increasing Decreasing

all an+ 1 all

Figure 2.5

There are four methods commonly used to prove that a sequence is mono-

tone. For example, any of the following methods will show that {an} is mono-

tone increasing:

(a) By subtracting successive terms, show that Vn EN, an+l - an ~ 0.

(b) If all an are positive, divide successive terms and show that an+l ~ l.

an

• (c) If f(x) =ax is differentiable, show that Vx ~ 1, f'(x) ~ 0. (We shall not
  use this method before Chapter 6 where derivatives are introduced.)

(d) Use mathematical induction to show that Vn EN, an:::; an+l·

Examples 2.5.2 (a) The sequence { ~} is strictly decreasing, since Vn EN,

1 1 n-(n+l) -1

0

n+l -n n(n+l) n(n+l) < ·

(b) The sequence { ~} is strictly increasing, since Vn E N,

4n+5

3(n+l) 1~ = 3n+3. 4n+5=12n^2 +27n+15> 1

4(n+1)+5 4n+5 4n+9 3n 12n^2 +27n ·

(!:, ( c) The sequence { ~: } is strictly increasing after the first three terms,

smce

3n+1 n3 3n3

-------> 1 whenever 3n^3 > n^3 + 3n^2 +3n+1, or

(n + 1) 3 3n n^3 + 3n^2 + 3n + 1

n[n(2n - 3) - 3] - 1 > 0, which is true when n ~ 3. D