Monotone Sequences

1. Monotone Sequences

2. Let {an} be a sequence. We say it’s
* increasing if an< an+1
Monotone Sequences

3. Let {an} be a sequence. We say it’s
* increasing if an< an+1
* non–decreasing if an ≤ an+1
Monotone Sequences

4. Let {an} be a sequence. We say it’s
* increasing if an< an+1
* non–decreasing if an ≤ an+1
* decreasing if an > an+1
* non–increasing an ≥ an+1 for all n’s.
Monotone Sequences

5. Let {an} be a sequence. We say it’s
* increasing if an< an+1
* non–decreasing if an ≤ an+1
* decreasing if an > an+1
* non–increasing an ≥ an+1 for all n’s.
Example A. 1, 2, 3, 4…. is an increasing sequence
1, 2, 3, 3, 3, ….is a non–decreasing sequence
–1, –2, –3, –4, …is a decreasing sequence
–1, –2, –3, –3, –3, ….is a non–increasing sequence
and 1, –2, 3, –4… is none of these.
Monotone Sequences

6. Let {an} be a sequence. We say it’s
* increasing if an< an+1
* non–decreasing if an ≤ an+1
* decreasing if an > an+1
* non–increasing an ≥ an+1 for all n’s.
Example A. 1, 2, 3, 4…. is an increasing sequence
1, 2, 3, 3, 3, ….is a non–decreasing sequence
–1, –2, –3, –4, …is a decreasing sequence
–1, –2, –3, –3, –3, ….is a non–increasing sequence
and 1, –2, 3, –4… is none of these.
We call these sequences monotone sequences -
i.e. its terms head in one direction and don’t go
backward.
Monotone Sequences

7. Let {an} be a sequence. We say it’s
* increasing if an< an+1
* non–decreasing if an ≤ an+1
* decreasing if an > an+1
* non–increasing an ≥ an+1 for all n’s.
Example A. 1, 2, 3, 4…. is an increasing sequence
1, 2, 3, 3, 3, ….is a non–decreasing sequence
–1, –2, –3, –4, …is a decreasing sequence
–1, –2, –3, –3, –3, ….is a non–increasing sequence
and 1, –2, 3, –4… is none of these.
We call these sequences monotone sequences -
i.e. its terms head in one direction and don’t go
backward. Increasing/decreasing sequences
are said to be strictly monotone.
Monotone Sequences

8. (Monotone-Tests)
Here are some basic methods for justifying that a
given sequence is monotone from its defining formula.
Monotone Sequences

9. (Monotone-Tests)
A sequence {an}n=1 is increasing if
I. (difference method) an+1 – an > 0 for all n
Here are some basic methods for justifying that a
given sequence is monotone from its defining formula.
Monotone Sequences

10. (Monotone-Tests)
A sequence {an}n=1 is increasing if
I. (difference method) an+1 – an > 0 for all n
II. (ratio method) an+1 / an or an+1 (1/an) > 1 for all n
Here are some basic methods for justifying that a
given sequence is monotone from its defining formula.
Monotone Sequences

11. (Monotone-Tests)
A sequence {an}n=1 is increasing if
I. (difference method) an+1 – an > 0 for all n
II. (ratio method) an+1 / an or an+1 (1/an) > 1 for all n
III. (derivative method) a'(x) > 0 for x > 0
Here are some basic methods for justifying that a
given sequence is monotone from its defining formula.
Monotone Sequences

12. (Monotone-Tests)
A sequence {an}n=1 is increasing if
I. (difference method) an+1 – an > 0 for all n
II. (ratio method) an+1 / an or an+1 (1/an) > 1 for all n
III. (derivative method) a'(x) > 0 for x > 0
Here are some basic methods for justifying that a
given sequence is monotone from its defining formula.
Monotone Sequences
Note that we may replace “>” in the theorem
* with “<” for decreasing sequences,

13. (Monotone-Tests)
A sequence {an}n=1 is increasing if
I. (difference method) an+1 – an > 0 for all n
II. (ratio method) an+1 / an or an+1 (1/an) > 1 for all n
III. (derivative method) a'(x) > 0 for x > 0
Here are some basic methods for justifying that a
given sequence is monotone from its defining formula.
Monotone Sequences
Note that we may replace “>” in the theorem
* with “<” for decreasing sequences,
* with “≥” for non–decreasing sequences.
* with “≤” for non–increasing sequences.

14. (Monotone-Tests)
A sequence {an}n=1 is increasing if
I. (difference method) an+1 – an > 0 for all n
II. (ratio method) an+1 / an or an+1 (1/an) > 1 for all n
III. (derivative method) a'(x) > 0 for x > 0
Here are some basic methods for justifying that a
given sequence is monotone from its defining formula.
Monotone Sequences
Note that we may replace “>” in the theorem
* with “<” for decreasing sequences,
* with “≥” for non–decreasing sequences.
* with “≤” for non–increasing sequences.
The main fact of a monotone sequence {an} is that
it either converges or it goes to ±∞.

15. (Monotone-Tests)
A sequence {an}n=1 is increasing if
I. (difference method) an+1 – an > 0 for all n
II. (ratio method) an+1 / an or an+1 (1/an) > 1 for all n
III. (derivative method) a'(x) > 0 for x > 0
Here are some basic methods for justifying that a
given sequence is monotone from its defining formula.
Monotone Sequences
Note that we may replace “>” in the theorem
* with “<” for decreasing sequences,
* with “≥” for non–decreasing sequences.
* with “≤” for non–increasing sequences.
The main fact of a monotone sequence {an} is that
it either converges or it goes to ±∞. Monotonicity
prevents the terms from behaving chaotically.

16. Example B. Justify that {an} = {2n/n!} is a decreasing
sequences for n = 2, 3, 4, ...
Monotone Sequences

17. Example B. Justify that {an} = {2n/n!} is a decreasing
sequences for n = 2, 3, 4, ...
The most suitable method is the ratio method.
Monotone Sequences

18. Example B. Justify that {an} = {2n/n!} is a decreasing
sequences for n = 2, 3, 4, ...
The most suitable method is the ratio method.
an+1 =
Monotone Sequences
n!
2n(n+1)!
2n+1
* =
an
1

19. Example B. Justify that {an} = {2n/n!} is a decreasing
sequences for n = 2, 3, 4, ...
The most suitable method is the ratio method.
an+1 =
Monotone Sequences
n!
2n(n+1)!
2n+1
* = 2
n+1
< 1 for n = 2, 3, …
an
1

20. Example B. Justify that {an} = {2n/n!} is a decreasing
sequences for n = 2, 3, 4, ...
The most suitable method is the ratio method.
an+1 =
Monotone Sequences
n!
2n(n+1)!
2n+1
* = 2
n+1
< 1 for n = 2, 3, …
an
1
So {2n/n!} is a decreasing sequence where n > 1.

21. Example B. Justify that {an} = {2n/n!} is a decreasing
sequences for n = 2, 3, 4, ...
The most suitable method is the ratio method.
The behavior of a sequence is determined by the
"tail" of the sequence.
an+1 =
Monotone Sequences
n!
2n(n+1)!
2n+1
* = 2
n+1
< 1 for n = 2, 3, …
an
1
So {2n/n!} is a decreasing sequence where n > 1.

22. Example B. Justify that {an} = {2n/n!} is a decreasing
sequences for n = 2, 3, 4, ...
The most suitable method is the ratio method.
The behavior of a sequence is determined by the
"tail" of the sequence. We say a sequence is
eventually monotone if the sequence is monotone
for all an’s where n > k for some k.
an+1 =
Monotone Sequences
n!
2n(n+1)!
2n+1
* = 2
n+1
< 1 for n = 2, 3, …
an
1
So {2n/n!} is a decreasing sequence where n > 1.

23. Example B. Justify that {an} = {2n/n!} is a decreasing
sequences for n = 2, 3, 4, ...
The most suitable method is the ratio method.
The behavior of a sequence is determined by the
"tail" of the sequence. We say a sequence is
eventually monotone if the sequence is monotone
for all an’s where n > k for some k.
1, 0, 2, 0, 3, 4, 5, 6, 7… is eventually increasing
because it’s increasing beyond the 3’rd term (k = 3).
an+1 =
Monotone Sequences
n!
2n(n+1)!
2n+1
* = 2
n+1
< 1 for n = 2, 3, …
an
1
So {2n/n!} is a decreasing sequence where n > 1.

24. Example B. Justify that {an} = {2n/n!} is a decreasing
sequences for n = 2, 3, 4, ...
The most suitable method is the ratio method.
The behavior of a sequence is determined by the
"tail" of the sequence. We say a sequence is
eventually monotone if the sequence is monotone
for all an’s where n > k for some k.
1, 0, 2, 0, 3, 4, 5, 6, 7… is eventually increasing
because it’s increasing beyond the 3’rd term (k = 3).
We also say it’s increasing “for sufficiently large n”
(in this case for n > 3).
an+1 =
Monotone Sequences
n!
2n(n+1)!
2n+1
* = 2
n+1
< 1 for n = 2, 3, …
an
1
So {2n/n!} is a decreasing sequence where n > 1.

25. Monotone Sequences
{an} is said to be bounded above (below) if there
exists a number A such that an≤ A (A ≤ an) for all an.

26. Monotone Sequences
{an} is said to be bounded above (below) if there
exists a number A such that an≤ A (A ≤ an) for all an.
A sequence {an} is said to be bounded if
it's bounded both above and below.

27. Monotone Sequences
{an} is said to be bounded above (below) if there
exists a number A such that an≤ A (A ≤ an) for all an.
A sequence {an} is said to be bounded if
it's bounded both above and below.
{sin(n)} = {sin(1)≈0.84, sin(2)≈0.90, sin(3)≈0.14, ..}
is bounded between –1 and 1,

28. Monotone Sequences
{an} is said to be bounded above (below) if there
exists a number A such that an≤ A (A ≤ an) for all an.
A sequence {an} is said to be bounded if
it's bounded both above and below.
{sin(n)} = {sin(1)≈0.84, sin(2)≈0.90, sin(3)≈0.14, ..}
is bounded between –1 and 1, but it oscillates,
so {sin(n)} diverges.

29. Monotone Sequences
{an} is said to be bounded above (below) if there
exists a number A such that an≤ A (A ≤ an) for all an.
A sequence {an} is said to be bounded if
it's bounded both above and below.
{sin(n)} = {sin(1)≈0.84, sin(2)≈0.90, sin(3)≈0.14, ..}
is bounded between –1 and 1, but it oscillates,
so {sin(n)} diverges.
However, bounded and eventually monotone
sequences do converge.

30. (Monotone-Sequence-Convergent Theorem)
Monotone Sequences
{an} is said to be bounded above (below) if there
exists a number A such that an≤ A (A ≤ an) for all an.
A sequence {an} is said to be bounded if
it's bounded both above and below.
{sin(n)} = {sin(1)≈0.84, sin(2)≈0.90, sin(3)≈0.14, ..}
is bounded between –1 and 1, but it oscillates,
so {sin(n)} diverges.
However, bounded and eventually monotone
sequences do converge.

31. (Monotone-Sequence-Convergent Theorem)
Let {an} be an eventually monotone sequence.
* If {an} is bounded, then it converges,
Monotone Sequences
{an} is said to be bounded above (below) if there
exists a number A such that an≤ A (A ≤ an) for all an.
A sequence {an} is said to be bounded if
it's bounded both above and below.
{sin(n)} = {sin(1)≈0.84, sin(2)≈0.90, sin(3)≈0.14, ..}
is bounded between –1 and 1, but it oscillates,
so {sin(n)} diverges.
However, bounded and eventually monotone
sequences do converge.

32. (Monotone-Sequence-Convergent Theorem)
Let {an} be an eventually monotone sequence.
* If {an} is bounded, then it converges,
i.e. lim an = N for some N as n∞.
Monotone Sequences
{an} is said to be bounded above (below) if there
exists a number A such that an≤ A (A ≤ an) for all an.
A sequence {an} is said to be bounded if
it's bounded both above and below.
{sin(n)} = {sin(1)≈0.84, sin(2)≈0.90, sin(3)≈0.14, ..}
is bounded between –1 and 1, but it oscillates,
so {sin(n)} diverges.
However, bounded and eventually monotone
sequences do converge.

33. (Monotone-Sequence-Convergent Theorem)
Let {an} be an eventually monotone sequence.
* If {an} is bounded, then it converges,
i.e. lim an = N for some N as n∞.
* If {an} is not bounded then {an} diverges to +∞ or –∞.
Monotone Sequences
{an} is said to be bounded above (below) if there
exists a number A such that an≤ A (A ≤ an) for all an.
A sequence {an} is said to be bounded if
it's bounded both above and below.
{sin(n)} = {sin(1)≈0.84, sin(2)≈0.90, sin(3)≈0.14, ..}
is bounded between –1 and 1, but it oscillates,
so {sin(n)} diverges.
However, bounded and eventually monotone
sequences do converge.

34. Monotone Sequences
Hence a bounded–below non–increasing sequence
is a CG sequence.
a lower
bound L

35. Monotone Sequences
Hence a bounded–below non–increasing sequence
is a CG sequence.
a lower
bound L
a1a2a3... a4

36. And a bounded–above non–decreasing sequence
is a CG sequence.
Monotone Sequences
an upper
bound U
Hence a bounded–below non–increasing sequence
is a CG sequence.
a lower
bound L
a1a2a3... a4

37. And a bounded–above non–decreasing sequence
is a CG sequence.
Monotone Sequences
an upper
bound U
a1 a2 a3 a4 . ..
Hence a bounded–below non–increasing sequence
is a CG sequence.
a lower
bound L
a1a2a3... a4

38. And a bounded–above non–decreasing sequence
is a CG sequence.
Monotone Sequences
an upper
bound U
a1 a2 a3 a4 . ..
Hence a bounded–below non–increasing sequence
is a CG sequence.
a lower
bound L
a1a2a3... a4
So a non–increasing positive sequence converges
because it’s bounded below by 0.

39. And a bounded–above non–decreasing sequence
is a CG sequence.
Monotone Sequences
an upper
bound U
a1 a2 a3 a4 . ..
Hence a bounded–below non–increasing sequence
is a CG sequence.
a lower
bound L
a1a2a3... a4
So a non–increasing positive sequence converges
because it’s bounded below by 0.
And a non–decreasing negative sequence converges
because it’s bounded above 0.

40. Monotone Sequences
Example C. Show that { } converges.2n
1 + 3n

41. Monotone Sequences
We will show that an+1 – an < 0:
Example C. Show that { } converges.2n
1 + 3n

42. Monotone Sequences
2n+1
1 + 3n+1
– 2n
1 + 3n
LCD = (1 + 3n+1)(1 + 3n)
We will show that an+1 – an < 0:
Example C. Show that { } converges.2n
1 + 3n

43. Monotone Sequences
2n+1
1 + 3n+1
– 2n
1 + 3n
=
LCD = (1 + 3n+1)(1 + 3n)
2n+1(1 + 3n) – 2n(1 + 3n+1) / LCD
We will show that an+1 – an < 0:
Example C. Show that { } converges.2n
1 + 3n

44. Monotone Sequences
2n+1
1 + 3n+1
– 2n
1 + 3n
=
LCD = (1 + 3n+1)(1 + 3n)
2n+1(1 + 3n) – 2n(1 + 3n+1) / LCD
= 2n[2(1 + 3n) – (1 + 3n+1)] / LCD
We will show that an+1 – an < 0:
Example C. Show that { } converges.2n
1 + 3n

45. Monotone Sequences
2n+1
1 + 3n+1
– 2n
1 + 3n
=
LCD = (1 + 3n+1)(1 + 3n)
2n+1(1 + 3n) – 2n(1 + 3n+1) / LCD
= 2n[2(1 + 3n) – (1 + 3n+1)] / LCD
= 2n[2 + 2*3n – 1 – 3n+1] / LCD
We will show that an+1 – an < 0:
Example C. Show that { } converges.2n
1 + 3n

46. Monotone Sequences
2n+1
1 + 3n+1
– 2n
1 + 3n
=
LCD = (1 + 3n+1)(1 + 3n)
2n+1(1 + 3n) – 2n(1 + 3n+1) / LCD
= 2n[2(1 + 3n) – (1 + 3n+1)] / LCD
= 2n[2 + 2*3n – 1 – 3n+1] / LCD
= 2n[1 + 3n (2 – 3)] / LCD
We will show that an+1 – an < 0:
Example C. Show that { } converges.2n
1 + 3n

47. Monotone Sequences
2n+1
1 + 3n+1
– 2n
1 + 3n
=
LCD = (1 + 3n+1)(1 + 3n)
2n+1(1 + 3n) – 2n(1 + 3n+1) / LCD
= 2n[2(1 + 3n) – (1 + 3n+1)] / LCD
= 2n[2 + 2*3n – 1 – 3n+1] / LCD
= 2n[1 + 3n (2 – 3)] / LCD
= 2n[1 – 3n ] / LCD < 0 for n = 1, 2, 3 ..
We will show that an+1 – an < 0:
Example C. Show that { } converges.2n
1 + 3n

48. Monotone Sequences
2n+1
1 + 3n+1
– 2n
1 + 3n
=
LCD = (1 + 3n+1)(1 + 3n)
2n+1(1 + 3n) – 2n(1 + 3n+1) / LCD
= 2n[2(1 + 3n) – (1 + 3n+1)] / LCD
= 2n[2 + 2*3n – 1 – 3n+1] / LCD
= 2n[1 + 3n (2 – 3)] / LCD
= 2n[1 – 3n ] / LCD < 0 for n = 1, 2, 3 ..
We will show that an+1 – an < 0:
So the sequence is decreasing and it’s bounded
below by 0, hence it’s a CG sequence.
Example C. Show that { } converges.2n
1 + 3n