00:00Hi guys, okay, in this
00:02lesson we're going to look
00:02at the McLaren series for
00:04this function 1 plus x
00:05to the power of P
00:06and this is called the
00:08binomial series and you will
00:11notice that this is where
00:13we get the formula for
00:15the binomial expansion, the general
00:18formula for the binomial expansion.
00:21Now first what I'm going
00:22to do is just this
00:25example here, the head square
00:26root of 1 plus x
00:27which is obviously
00:28see 1 plus x to
00:28the power of a half.
00:30So this is like p
00:31can be, so p is
00:33an element of the rational
00:37number. So p can be
00:39any rational number positive or
00:41negative. So first I just
00:43want to do this example
00:45normally using our McLaren series.
00:47So same way we've got
00:49f of x, we need,
00:50so f of x equals
00:531 plus x to the
00:55power of a half.
00:56F dash of x, and
01:01it says the first four
01:03non -zero terms. So I'm
01:07going to need the first
01:08three derivatives. So F dash
01:11of x is a half
01:12times one plus x to
01:15the minus half. F double
01:19dash of x is negative
01:21a quarter, one plus x,
01:24to the minus 3 over
01:262. And the third derivative
01:29is positive 3 over 8,
01:351 plus x, to the
01:38power of negative 5 over
01:402. Now, I need f
01:42of 0. f of 0
01:44is, because I'm going to
01:46sub in 0 here, so
01:47it's 1. f dash of
01:500 is,
01:52Well look, all of the,
01:53when I sub in zero,
01:55all of these are going
01:55to turn into one. So
01:57that's going to be one.
01:59That's going to be one.
01:59So f dash of zero
02:00here is a half. And
02:05f double dash of zero
02:07is negative a quarter. And
02:10the third derivative at zero
02:13is three -eighths. Now we
02:16just need to sub these
02:18into this. So f of
02:19x
02:20I'm going to do approximately
02:22equal to f of x
02:24is approximately equal to one
02:26plus this times x, so
02:28it's a half x plus
02:31this times x squared over
02:33two factorial. So minus this
02:36times x squared over two
02:39factorial. And then plus this
02:44times x cubed over three
02:47factorial.
02:48will simplify this into one
02:51plus a half x minus
02:54an eighth x squared plus
02:58this would be one of
03:00the three's with cancelable f
03:02of two times eight. So
03:03a 16th x cubed. And
03:06so this is the first
03:08four terms. First four non
03:10zero terms, non zero terms
03:12is just a fancy way
03:13of saying terms, but aren't
03:14zero because if you had
03:15a zero term
03:16I've explained that in previous
03:18videos. So it's the first
03:22four terms of the McLaren
03:23series for one, the square
03:25root of one plus x.
03:27Now sometimes you might get
03:28a question that says, use
03:30this expansion to estimate a
03:34value for root two. And
03:39if that's the case, what
03:40you need to do is
03:41sub one into here.
03:44and you'll get an estimate
03:47for root two because root
03:49one plus one obviously gives
03:51me the root two, so
03:52x is one, sub it
03:54in there and you get
03:55a value for root two.
03:57Okay, that's the first bit
03:58of the lesson. The second
03:59bit is I want to
04:01do the same kind of
04:02method to come up with
04:04a general formula for this
04:06one plus x to the
04:07power of p. And we
04:11are, yeah, certainly we're gonna
04:12use this for the binomial
04:15expansion. But also, this is
04:18a useful, once you know
04:19this general formula, you can
04:21use it to find any
04:25McLaren series for expression or
04:31function in this form. So
04:32same thing. I'm going to
04:33start with f of x,
04:36f of x equals f
04:39of x equals
04:40one plus x to the
04:43p. F dash of x
04:46equals p, so it's the
04:48same thing as p times
04:49this, one plus x to
04:53the p minus one, so
04:54I subtract one, then F
04:57double dash of x. Now
04:59where do I stop? Well,
05:02he doesn't say here, but
05:03I just want to do
05:04enough, so I see the
05:06pattern, what's going on?
05:08I've double dash of x
05:10is p times p minus
05:131, so I'm going to
05:14multiply over this power times
05:161 plus x to the
05:18power of p minus 2.
05:20Then, f triple dash or
05:24the third derivative is p
05:26times p minus 1 times
05:29p minus 2 times 1
05:33plus x to the power
05:36of p
05:36minus 3. Now, hopefully you
05:40can start to see the
05:41pattern, get the fourth derivative,
05:42I just times the by
05:43p minus 3, and this
05:44will become to the power
05:45of p minus 4. Now,
05:47similarly to the first example,
05:49because I want f of
05:510 and f dash of
05:520, I want all this
05:53at 0, this kind of
05:56term, whatever its power is,
05:58it's 1 plus 0, so
05:59it's just 1 to the
06:00power of whatever is going
06:02to give me the 1.
06:03So f dash of
06:040 is 1, sorry, f
06:07of 0 is 1, f
06:08dash of 0 is just
06:11p times 1, which is
06:13also 1. Sorry, what am
06:15I saying? P times 1
06:16is p. Second derivative, f
06:20dash, double dash of 0
06:22is equal to p times
06:24p minus 1 times 1,
06:26so it's p times p
06:29minus 1, p
06:33e times p minus 1
06:35times 1, which is just
06:36that. And then here I
06:38have p dash dash dash
06:41of 0 equals, so the
06:44third derivative is p times
06:47p minus 1 times p
06:50minus 2 times 1. Okay,
06:53now the McLaren series, what
06:55is it? Well, it is
06:58f of 0. So that's
06:591 plus
07:01f dash of zero, x,
07:05which is p, x plus,
07:09and second derivative, p times
07:12p minus one, x squared
07:14over, I need to move
07:17this down, x squared over,
07:24over two factorial, I'll just
07:28put it in the
07:29like this over two factorial
07:31plus p times p minus
07:35one, p times p minus
07:37one times p minus two
07:40over three factorial, et cetera.
07:45And what we get is,
07:47so I want to get
07:48my, I want to get
07:49the kind of general term
07:50here. And what I get
07:52is I'll do a plus
07:54dot dot dot dot dot
07:57dot plus. So this keeps
07:59going. And then I get,
08:01so I'm gonna get P
08:04times P minus one, times
08:07P minus two. All the
08:11way down to P minus,
08:18this is a little bit
08:18tricky, P minus N minus
08:20one, P times N minus
08:23one.
08:25Sorry, this is x, this
08:26should have been x cubed.
08:29And then this is x
08:30to the n, and it's
08:32all over. This is all
08:35over. This is one term
08:36here, and this is all
08:37over n factorial, and then
08:41that kind of just continues.
08:46So, let's try and get
08:48our head around this a
08:49second. Let's have a look.
08:52So the first, hopefully the
08:53first
08:53first, second, third terms, fourth
08:56terms all make sense, because
08:58you've seen it from the
08:59derivative. Now what happens is
09:01here, look here I've x
09:02squared over two factorial, x
09:04cubed over three factorial, x
09:06to the four over four
09:07factorial, so x to the
09:08n over n factorial, that
09:10makes sense for the general
09:11term. But what happens with
09:13where do I stop with
09:14the P's? Well, when it's
09:16two factorial, I stop at
09:17P minus one. When it's
09:19three factorial, I stop at
09:20P minus two.
09:21When it's n factorial, I
09:23stop at n minus 1.
09:25And you can write this
09:27and you might see this
09:29written in exam, sort of,
09:31in books or whatever. You
09:32certainly can write this as
09:33p minus n plus 1.
09:36And without the bracket, it's
09:37the same thing. So that
09:39is when I said find
09:41the McClure and series for
09:42this, for p is an
09:43element of rational numbers. This
09:46is what I want. I
09:47wanted the general, the general,
09:49Maclaurin series, which is called
09:53the binomial series. And this
09:56is when you look at
09:57your formula in your formula
09:59booklet for the binomial theorem,
10:04this is where it comes
10:06from. And I think when
10:07I did that lesson, I
10:09said I would explain where
10:11this comes from in one
10:15of my Maclaurin series lessons.
10:17This is that lesson. Okay,
10:19hopefully that makes sense. Yeah,
10:23I'll talk to you in
10:25the next lesson.