1
00:00:00,008 --> 00:00:01,640
Alternating series.

2
00:00:01,640 --> 00:00:07,984
[MUSIC]

3
00:00:07,984 --> 00:00:13,842
What is an alternating series?
For example, this series the sum

4
00:00:13,842 --> 00:00:19,670
n equals 1 to infinity of minus 1 to the n
plus 1 over n,

5
00:00:19,670 --> 00:00:25,240
is an example of an alternating series.
Chris, what does that even mean?

6
00:00:25,240 --> 00:00:26,230
Let me write down some of the terms,

7
00:00:26,230 --> 00:00:27,560
because I think will be a little bit
clearer.

8
00:00:27,560 --> 00:00:32,380
So if I plug in n equals 1, I get negative
1 squared, which is 1 over 1.

9
00:00:32,380 --> 00:00:33,400
So 1 over 1.

10
00:00:33,400 --> 00:00:37,477
When I plug in n equals 2, I get negative
1 cubed over 2.

11
00:00:37,477 --> 00:00:41,530
That's negative 1 over 2, so minus a half.

12
00:00:41,530 --> 00:00:47,813
When I plug in n equals 3, that's negative
1 to the 4th over 3.

13
00:00:47,813 --> 00:00:48,678
That's 1 over 3.

14
00:00:48,678 --> 00:00:49,682
That's plus a 3rd.

15
00:00:49,682 --> 00:00:52,532
When I plug in n equals 4, I get negative
1 to

16
00:00:52,532 --> 00:00:56,215
the 5th, which is negative 1 over 4, so
minus 1 4th.

17
00:00:56,215 --> 00:00:58,503
When I plug in n equals 5,

18
00:00:58,503 --> 00:01:04,300
I get negative 1 to the 6th over 5.
That's 1 over 5, so plus a 5th.

19
00:01:04,300 --> 00:01:08,305
When I plug in n equals 6, I get negative
1 to the

20
00:01:08,305 --> 00:01:13,550
7th over 6 so, that's just negative 1 over
6, so minus a 6th.

21
00:01:13,550 --> 00:01:15,378
And then it's going to keep on going like
that.

22
00:01:15,378 --> 00:01:18,218
But I'm flip-flopping between these two
colors.

23
00:01:18,218 --> 00:01:19,595
I'm alternating sines.

24
00:01:19,595 --> 00:01:19,970
Right?

25
00:01:19,970 --> 00:01:23,810
I'm adding, subtracting, adding,
subtracting, adding, subtracting as

26
00:01:23,810 --> 00:01:27,705
an alternating series.
So that's what I mean by alternating.

27
00:01:27,705 --> 00:01:33,164
Well here's a precise definition.
The series is sum a sub

28
00:01:33,164 --> 00:01:38,708
n, and goes from 1 to infinity,

29
00:01:38,708 --> 00:01:44,450
is called an alternating series,

30
00:01:44,450 --> 00:01:49,202
if a sub n is equal to negative

31
00:01:49,202 --> 00:01:55,142
1 to the nth power times b sub n, and

32
00:01:55,142 --> 00:02:00,820
all the b sub n are the same sign.

33
00:02:00,820 --> 00:02:04,890
So maybe the b sub n sequence, is say all
positive, and

34
00:02:04,890 --> 00:02:08,900
then a sub n has this term here, times a
positive sequence.

35
00:02:08,900 --> 00:02:13,590
Well this term is what makes the signs,
the s i g ns, flip flop back and forth.

36
00:02:14,620 --> 00:02:16,130
Well here's a bit of a warning.

37
00:02:16,130 --> 00:02:18,480
Not every series which has both positive

38
00:02:18,480 --> 00:02:21,980
and negative terms, is an alternating
series.

39
00:02:21,980 --> 00:02:26,941
For example this series the sum n goes
from 1

40
00:02:26,941 --> 00:02:32,100
to infinity of minus 1 to the n times sign
n over n.

41
00:02:32,100 --> 00:02:34,027
This is not an alternating series.

42
00:02:34,027 --> 00:02:36,000
I mean yes, it's got the minus 1 to the

43
00:02:36,000 --> 00:02:39,650
n here, but the sine of n also introduces
it's own

44
00:02:39,650 --> 00:02:43,270
quite complicated pattern of positive and
negative terms.

45
00:02:43,270 --> 00:02:45,065
So this is a series, some of the terms are

46
00:02:45,065 --> 00:02:47,940
positive, some of the terms are negative,
but if not alternating.

47
00:02:47,940 --> 00:02:52,071
Alright, this is not an alternating series
you can cook

48
00:02:52,071 --> 00:02:55,770
up other examples sum n goes from 1 to
infinity.

49
00:02:55,770 --> 00:03:00,260
Say, minus 1 to the n times n plus 1 over
2.

50
00:03:00,260 --> 00:03:02,370
So that's all in the exponent there.

51
00:03:02,370 --> 00:03:04,512
n times n plus 1 over 2,

52
00:03:04,512 --> 00:03:06,720
say all over n.

53
00:03:06,720 --> 00:03:10,990
This is also not an alternating series,
and it's still got a minus 1

54
00:03:10,990 --> 00:03:15,020
there, to raise to some power, but the
power is a little bit more complicated.

55
00:03:15,020 --> 00:03:17,818
n times n plus 1 over 2.

56
00:03:17,818 --> 00:03:21,414
terms are some of em will be positive,
some are going to be negative,

57
00:03:21,414 --> 00:03:24,980
but they're not flip-flopping in sign, in
s i g n back and forth.

58
00:03:26,310 --> 00:03:29,938
In contrast, here's an example that is an
alternating series.

59
00:03:29,938 --> 00:03:33,678
the sum n goes from 1 to infinity, say of

60
00:03:33,678 --> 00:03:38,510
minus 1 to the n times sine squared n over
n.

61
00:03:38,510 --> 00:03:42,350
Sine squared n, this turns out to always
be positive.

62
00:03:42,350 --> 00:03:42,840
Right?

63
00:03:42,840 --> 00:03:44,900
The n is positive here, at minus 1 to the
n.

64
00:03:44,900 --> 00:03:49,150
This is the only thing that's affecting
the s i g n of this

65
00:03:49,150 --> 00:03:52,050
term, and this does indeed then flip-flop

66
00:03:52,050 --> 00:03:54,750
back and forth, between negative,
positive, negative,

67
00:03:54,750 --> 00:04:02,384
positive, negative, positive.
This is an alternating series.

68
00:04:02,384 --> 00:04:09,584
[SOUND]