1
00:00:00,270 --> 00:00:03,816
Welcome to week five of sequences and
series.

2
00:00:03,816 --> 00:00:10,240
[MUSIC]

3
00:00:10,240 --> 00:00:15,900
We're going to be looking at power series.
These are series that look like this.

4
00:00:15,900 --> 00:00:19,840
The sum, say, n goes from zero to infinity
of

5
00:00:19,840 --> 00:00:25,300
a sub n, just some numbers, times x to the
n.

6
00:00:25,300 --> 00:00:29,010
That means you get to pick a sequence a
sub n.

7
00:00:29,010 --> 00:00:35,280
For example maybe a sub n is 2 to the n,
just the sequence of the powers

8
00:00:35,280 --> 00:00:36,000
of two.

9
00:00:36,000 --> 00:00:41,140
The important thing here is that a sub n
doesn't depend on x in any way.

10
00:00:41,140 --> 00:00:46,340
A sub n is just a formula in this case
given in terms of n, but not x.

11
00:00:46,340 --> 00:00:49,950
And from that sequence, you build the
power series.

12
00:00:49,950 --> 00:00:54,510
Well continuing with this example, if a
sub n is 2 to the n, then the associated

13
00:00:54,510 --> 00:01:01,040
power series is the sum n goes from zero
to infinity of 2 to the n times

14
00:01:01,040 --> 00:01:02,770
x to the n.

15
00:01:02,770 --> 00:01:06,200
The cool thing is that in a ton of cases
the power

16
00:01:06,200 --> 00:01:08,570
series that we're building are actually

17
00:01:08,570 --> 00:01:11,500
representing functions we already know
about.

18
00:01:11,500 --> 00:01:12,560
Well, what about this case?

19
00:01:12,560 --> 00:01:13,740
What happens here?

20
00:01:13,740 --> 00:01:22,340
Well, one over one minus x is the sum n
goes from zero to infinity of x to the n.

21
00:01:22,340 --> 00:01:25,289
That's just the formula for summing a
geometric series.

22
00:01:26,450 --> 00:01:30,065
Now look at what happens if I replace x by
2 to the n.

23
00:01:30,065 --> 00:01:30,310
[INAUDIBLE]

24
00:01:30,310 --> 00:01:35,930
1 over 1 minus 2x.
That must be the sum n goes from

25
00:01:35,930 --> 00:01:42,070
zero to inifinity of 2x to the n.
Which is exactly what I've got here.

26
00:01:42,070 --> 00:01:47,720
This is the sum n goes from zero to
infinity of two to the n times x to the n.

27
00:01:47,720 --> 00:01:51,920
So this mysterious seeming power series is
actually just a complicated

28
00:01:51,920 --> 00:01:55,685
way of writing down this very reasonable
seeming function, 1 over

29
00:01:55,685 --> 00:01:57,760
1 minus 2x.

30
00:01:57,760 --> 00:02:02,200
The other cool thing is that power series
are like polynomials.

31
00:02:02,200 --> 00:02:04,490
If I just write down the first few terms,
right, I

32
00:02:04,490 --> 00:02:06,660
could just look at the first few terms of
this series.

33
00:02:06,660 --> 00:02:13,550
It's 1 plus 2x plus 4x squared plus 8x
cubed.

34
00:02:13,550 --> 00:02:16,950
If I truncate this series, I just get a
polynomial.

35
00:02:16,950 --> 00:02:18,840
And it is as soon as I write the plus

36
00:02:18,840 --> 00:02:21,240
dot dot dot, as soon as I'm thinking of
this as

37
00:02:21,240 --> 00:02:23,070
sort of a polynomial that goes on forever,

38
00:02:23,070 --> 00:02:25,590
well that's really what a, a power series
is.

39
00:02:27,040 --> 00:02:29,250
So power series are really very cool.

40
00:02:29,250 --> 00:02:31,460
They let us translate a lot of our

41
00:02:31,460 --> 00:02:36,160
intuition for polynomials into other, more
complicated functions.

42
00:02:36,160 --> 00:02:37,460
We're going to see that there are

43
00:02:37,460 --> 00:02:40,900
power series representations from very
complicated functions.

44
00:02:40,900 --> 00:02:42,585
Like sine and cosine.

45
00:02:42,585 --> 00:02:46,380
But, since these power series look like
polynomials, its going to

46
00:02:46,380 --> 00:02:48,657
let us translate some of that intuition

47
00:02:48,657 --> 00:02:53,082
about polynomials into those more
complicated transindental functions.

48
00:02:53,082 --> 00:03:03,082
[SOUND]