1
00:00:00,300 --> 00:00:02,170
The ratio test is awesome.

2
00:00:08,860 --> 00:00:10,560
What test should I apply?

3
00:00:10,560 --> 00:00:14,380
Well, for this series, the ratio test will
work wonderfully.

4
00:00:14,380 --> 00:00:18,598
I can really tell that the ratio test is
just going to be great for this.

5
00:00:18,598 --> 00:00:22,300
Because I've got these factorials and
these powers, so

6
00:00:22,300 --> 00:00:25,210
I can expect a lot of cancellation to
happen.

7
00:00:25,210 --> 00:00:29,160
Let's compute the limit of the ratio of
neighboring terms.

8
00:00:29,160 --> 00:00:34,110
So I'll write a sub n is n factorial over
n

9
00:00:34,110 --> 00:00:37,940
to the n, and I'm trying to calculate the
limit, as n

10
00:00:37,940 --> 00:00:43,710
approaches infinity of a sub n plus one
over a sub n.

11
00:00:43,710 --> 00:00:46,600
And that's the limit as n approaches
infinity.

12
00:00:46,600 --> 00:00:48,070
What's a sub n plus one?

13
00:00:48,070 --> 00:00:51,010
I just gotta replace these n's with n plus
one.

14
00:00:51,010 --> 00:00:59,240
That's n plus one factorial divided by n
plus one to the n plus oneth power divided

15
00:00:59,240 --> 00:01:01,480
by what's a sub n.

16
00:01:01,480 --> 00:01:05,840
Well, that's just n factorial over n to
the nth power.

17
00:01:05,840 --> 00:01:07,550
That can be simplified.

18
00:01:07,550 --> 00:01:10,250
First of all, I've got a fraction with
fractions in the

19
00:01:10,250 --> 00:01:13,490
numerator and denominator, so I can clean
that up a bit.

20
00:01:13,490 --> 00:01:17,200
This is the limit as n approaches
infinity.

21
00:01:17,200 --> 00:01:25,120
Of n plus one factorial times n to the n,
divided by n factorial

22
00:01:25,120 --> 00:01:29,430
times n plus one to the n plus oneth
power.

23
00:01:30,820 --> 00:01:31,880
Now what can I do?

24
00:01:31,880 --> 00:01:35,280
Well, I've got an n plus one factorial in
numerator, and an n factorial in

25
00:01:35,280 --> 00:01:39,060
the denominator, so this n factorial
cancels everything

26
00:01:39,060 --> 00:01:41,940
except for the n plus one term here.

27
00:01:41,940 --> 00:01:50,270
So this is the limit as n approaches
infinity of just n plus one

28
00:01:50,270 --> 00:01:51,800
times n to the n.

29
00:01:51,800 --> 00:01:57,760
Divided by, so that n factorial's gone
now, n plus one to the n plus oneth power.

30
00:01:59,170 --> 00:02:02,880
Well I've got an n plus one on the
numerator, and the power of n plus one in

31
00:02:02,880 --> 00:02:06,170
the denominator, so I can use this to
change

32
00:02:06,170 --> 00:02:08,750
this n plus one in just an nth power.

33
00:02:08,750 --> 00:02:15,810
So this is the limit as n approaches
infinity of n to the n over n

34
00:02:15,810 --> 00:02:17,560
plus one to the n.

35
00:02:37,650 --> 00:02:38,990
And if you like, I can just rewrite this a
bit too or analyze this limit.

36
00:02:38,990 --> 00:02:43,640
If you really love l'Hôpital's rule, you
could just apply l'Hôpital's rule.

37
00:02:43,640 --> 00:02:46,450
I don't really like l'Hôpital's rule that
much.

38
00:02:46,450 --> 00:02:49,310
So instead, I'm just going to recall a
useful fact.

39
00:02:49,310 --> 00:02:52,780
In fact, this might've been how you define
the number e.

40
00:02:52,780 --> 00:02:59,740
The limit, as n approaches infinity, of
one plus one over n to the nth power is e.

41
00:02:59,740 --> 00:03:02,900
Now, how can I take this fact and say
something about this limit?

42
00:03:03,910 --> 00:03:07,190
Well, I could combine this into a single
fraction.

43
00:03:07,190 --> 00:03:10,358
So one plus one over n is n over n plus
one

44
00:03:10,358 --> 00:03:15,300
over n, which means the limit of n plus
one over n.

45
00:03:15,300 --> 00:03:18,570
To the nth power as n approaches infinity
is e.

46
00:03:18,570 --> 00:03:24,030
And now this looks a whole lot like this.
And indeed all I have to do,

47
00:03:24,030 --> 00:03:28,910
is use the fact that the limit of a
reciprocal is the reciprocal of

48
00:03:28,910 --> 00:03:32,670
the limit to conclude that the limit of n
over n

49
00:03:32,670 --> 00:03:37,830
plus one to the nth power is in fact one
over e.

50
00:03:37,830 --> 00:03:41,096
What does that imply about the original
series?

51
00:03:41,096 --> 00:03:47,710
Now, one over e is less than one, and that
means that, according to the

52
00:03:47,710 --> 00:03:54,570
ratio test, the given series converges.
We can do even better.

53
00:03:54,570 --> 00:03:58,380
Does the series, n goes from one to
infinity of n factorial

54
00:03:58,380 --> 00:04:05,120
divided by n over two, to the nth power,
converge or diverge?

55
00:04:05,120 --> 00:04:08,740
Yes, this series converges.

56
00:04:08,740 --> 00:04:10,010
Let's see why.

57
00:04:10,010 --> 00:04:10,790
Well here we go.

58
00:04:10,790 --> 00:04:18,000
Let's set a sub n equal to n factorial
over N over two to the n.

59
00:04:18,000 --> 00:04:20,270
And my claim

60
00:04:20,270 --> 00:04:27,050
is that the sum, n goes from one to
infinity of a, sub n converges.

61
00:04:27,050 --> 00:04:30,120
To justify this claim, I'm going to use
the ratio test.

62
00:04:30,120 --> 00:04:34,520
So, big L, which is the limit as n
approaches

63
00:04:34,520 --> 00:04:40,260
infinity of a sub n plus one, over a sub
n.

64
00:04:40,260 --> 00:04:41,570
Well in this case what is that?

65
00:04:41,570 --> 00:04:45,408
That's the limit as n approaches infinity
of

66
00:04:45,408 --> 00:04:48,370
this with n replaced by n plus one.

67
00:04:48,370 --> 00:04:54,620
It's n plus one factorial over n plus one,
over two to the n plus one power.

68
00:04:54,620 --> 00:05:00,650
Divided by a sub n which is N factorial
over N over two to the nth power.

69
00:05:00,650 --> 00:05:04,240
This is kind of a mess because I've got
fractions, a numerator, and a denominator.

70
00:05:04,240 --> 00:05:07,290
So I can simplify that, can rewrite that
as the

71
00:05:07,290 --> 00:05:11,450
limit N goes to infinity of N plus one
factorial

72
00:05:11,450 --> 00:05:16,190
times n over two to the N divided by N

73
00:05:16,190 --> 00:05:22,610
factorial times N plus one over two to the
N plus one power.

74
00:05:22,610 --> 00:05:25,980
I've got an N plus one factorial divided
by an N factorial.

75
00:05:25,980 --> 00:05:30,530
Most of those terms cancel except for the
N plus one.

76
00:05:30,530 --> 00:05:33,460
So, I can rewrite that as just N plus one
on the numerator.

77
00:05:33,460 --> 00:05:35,480
Let me simplify this a bit too, or at
least let's expand it out.

78
00:05:35,480 --> 00:05:36,460
I can write this as n to the

79
00:05:36,460 --> 00:05:37,850
n, divided by two to the n.

80
00:05:37,850 --> 00:05:41,390
So it's n to the n, divided by two to the
n.

81
00:05:41,390 --> 00:05:45,540
And the denominator here, well the n
factorial goes away but I can rewrite this

82
00:05:45,540 --> 00:05:53,140
as n plus n to the n plus one power,
divided by two to the n plus one.

83
00:05:53,140 --> 00:05:55,430
Now I can keep simplifying this.

84
00:05:55,430 --> 00:05:57,840
I've got an an n plus one in the
numerator, an n plus one to the

85
00:05:57,840 --> 00:06:01,520
n plus one power in the denominator, I can
cancel one of those n plus ones in

86
00:06:01,520 --> 00:06:02,830
the demoninator.

87
00:06:02,830 --> 00:06:06,150
So now I've just got n plus one to the nth
power in the denominator.

88
00:06:06,150 --> 00:06:08,660
And in the numerator, I've still got n to
the n.

89
00:06:09,860 --> 00:06:12,500
And the numerator I'm dividing by two to
the n.

90
00:06:12,500 --> 00:06:14,130
So I can put that in the denominator.

91
00:06:14,130 --> 00:06:16,740
And in the denominator, I am diving by the
two to

92
00:06:16,740 --> 00:06:19,608
the n plus one so I can put that in the
numerator.

93
00:06:19,608 --> 00:06:22,946
I've got two to the n plus one divided by
two to the n.

94
00:06:22,946 --> 00:06:27,380
Everything except for a single factor of
two cancels.

95
00:06:27,380 --> 00:06:32,880
So what I'm left with here is, N to the N,
over N plus one to the N,

96
00:06:32,880 --> 00:06:38,280
times two.
But this, I can combine to be the limit.

97
00:06:38,280 --> 00:06:41,600
N goes to infinity of N over N plus one to
the N.

98
00:06:41,600 --> 00:06:43,850
And that's to get that times two.

99
00:06:43,850 --> 00:06:47,140
But we already started that, the limit of
this N over N

100
00:06:47,140 --> 00:06:51,140
plus one to the N, as N approaches
infinity, is one over E.

101
00:06:51,140 --> 00:06:52,492
So, this whole

102
00:06:52,492 --> 00:06:57,860
limit, is two over e and two over e is
less than one.

103
00:06:57,860 --> 00:07:02,710
So, by the ratio test, this series
converges.

104
00:07:02,710 --> 00:07:05,950
What if that two became a three?

105
00:07:05,950 --> 00:07:11,650
Does the series n goes from one to
infinity of n factorial over n divided

106
00:07:11,650 --> 00:07:17,700
by three to the nth power.
Converge or diverge?

107
00:07:17,700 --> 00:07:22,920
Now, this series doesn't converge.
Here's the argument that we used to

108
00:07:22,920 --> 00:07:29,260
show that the sum of n factorial over n
over two to the n converges.

109
00:07:29,260 --> 00:07:31,620
Now we switched that two for a three.

110
00:07:31,620 --> 00:07:36,160
We just figure out how this argument needs
to be changed.

111
00:07:36,160 --> 00:07:39,310
So let's replace this two here with a
three.

112
00:07:39,310 --> 00:07:43,280
And now the claim is that that series
doesn't converge

113
00:07:43,280 --> 00:07:46,260
anymore, but that it diverges.

114
00:07:46,260 --> 00:07:49,140
Again, I should be applying the ratio test
here, so I

115
00:07:49,140 --> 00:07:52,214
am looking at the limit of the ratio of
subsequent terms.

116
00:07:52,214 --> 00:07:58,930
But this has some twos that I swapped out
for threes.

117
00:07:58,930 --> 00:08:03,506
Here I've got some twos that I need to
swap out for threes.

118
00:08:03,506 --> 00:08:05,640
And here I got some twos.

119
00:08:05,640 --> 00:08:08,051
That I need to swap out for threes.

120
00:08:08,051 --> 00:08:12,460
And there's some more twos that need to be
replaced with threes.

121
00:08:12,460 --> 00:08:16,650
And here we got three to the n plus one
over three to the n.

122
00:08:16,650 --> 00:08:21,621
So instead of multiplying by two, I'm now
multiplying by three.

123
00:08:21,621 --> 00:08:23,259
This two becomes a three.

124
00:08:23,259 --> 00:08:27,060
And here's the worst part.
This two becomes a three.

125
00:08:27,060 --> 00:08:33,550
And three over e is not less than one.
Three over e is bigger than one.

126
00:08:33,550 --> 00:08:40,310
And because big L is bigger than one, the
ratio test says that this series diverges.

127
00:08:40,310 --> 00:08:42,860
Let me leave you, with a question.

128
00:08:42,860 --> 00:08:48,050
Does the series, n goes from one to
infinity of n factorial divided by n

129
00:08:48,050 --> 00:08:53,475
over e to the nth power converge or
diverge?

130
00:08:53,475 --> 00:09:01,709
[SOUND]

131
00:09:01,709 --> 00:09:05,210
[SOUND]