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PROFESSOR: So,
Professor Jerison is
relaxing in sunny
London, Ontario today
and sent me in as
his substitute again.
I'm glad to the here
and see you all again.
So our agenda today: he
said that he'd already
talked about power series
and Taylor's formula,
I guess on last week
right, on Friday?
So I'm going to go a
little further with that
and show you some examples,
show you some applications,
and then I have this
course evaluation survey
that I'll hand out in the last
10 minutes or so of the class.
I also have this handout
that he made that says
18.01 end of term 2007.
If you didn't pick this up
coming in, grab it going out.
People tend not to pick it
up when they walk in, I see.
So grab this when
you're going out.
There's some things
missing from it.
He has not decided
when his office hours
will be at the end of term.
He will have them, just
hasn't decided when.
So, check the website
for that information.
And we're looking forward to
the final exam, which is uh --
aren't we?
Any questions about
this technical stuff?
All right, let's talk about
power series for a little bit.
So I thought I should
review for you what
the story with power series is.
OK, could I have your
attention please?
So, power series is a way of
writing a function as a sum
of integral powers of x.
These a_0, a_1, and
so on, are numbers.
An example of a power
series is a polynomial.
Not to be forgotten,
one type of power series
is one which goes on for
a finite number of terms
and then ends, so that all of
the other, all the higher a_i's
are all 0.
This is a perfectly good
example of a power series;
it's a very special
kind of power series.
And part of what I
want to tell you today
is that power series
behave, almost exactly like,
polynomials.
There's just one
thing that you have
to be careful about when you're
using power series that isn't
a concern for polynomials,
and I'll show you
what that is in a minute.
So, you should think of them
as generalized polynomials.
The one thing that you
have to be careful about
is that there is a
number-- So one caution.
There's a number which I'll
call R, where R can be between 0
and it can also be infinity.
It's a number between 0
and infinity, inclusive,
so that when the absolute
value of x is less than R.
So when x is smaller than R
in size, the sum converges.
This sum-- that sum
converges to a finite value.
And when x is bigger
than R in absolute value,
the sum diverges.
This R is called the
radius of convergence.
So we'll see some examples of
what the radius of convergence
is in various powers series as
well, and how you find it also.
But, let me go on and
give you a few more
of the properties
about power series
which I think that professor
Jerison talked about earlier.
So one of them is there's
a radius of convergence.
Here's another one.
If you're inside of
the radius convergence,
then the function has
all its derivatives,
has all its derivatives,
just like a polynomial does.
You can differentiate
it over and over again.
And in terms of
those derivatives,
the number a_n in
the power series
can be expressed in terms of the
value of the derivative at 0.
And this is called
Taylor's formula.
So I'm saying that inside of
this radius of convergence,
the function that we're
looking at, this f(x),
can be written as the value of
the function at 0, that's a_0,
plus the value of
the derivative.
This bracket n means you
take the derivative n times.
So when n is 1, you take
the derivative once at 0,
divided by 1!, which is
!, and multiply it by x.
That's the linear term
in the power series.
And then the quadratic term is
you take the second derivative.
Remember to divide
by 2!, which is 2.
Multiply that by
x^2 and so on out.
So, in terms-- So
the coefficients
in the power series just record
the values of the derivatives
of the function at x = 0.
They can be computed
that way also.
Let's see.
I think that's the end
of my summary of things
that he talked about.
I think he did one
example, and I'll repeat
that example of a power series.
This example wasn't
due to David Jerison;
it was due to Leonard Euler.
It's the example of where the
function is the exponential
function e^x.
So, let's see.
Let's compute what-- I will just
repeat for you the computation
of the power series for
e^x, just because it's such
an important thing to do.
So, in order to do that, I have
to know what the derivative
of e^x is, and what the
second derivative of e^x is,
and so on, because that
comes into the Taylor formula
for the coefficients.
But we know what the derivative
of e^x is, it's just e^x again,
and it's that way
all the way down.
All the derivatives are
e^x over and over again.
So when I evaluate this at x =
0, well, the value of e^x is 1,
the value of e^x is 1 at x = 0.
You get a value of
1 all the way down.
So all these derivatives
at 0 have the value 1.
And now, when I plug
into this formula,
I find e^x is 1 plus 1*x
plus 1/2! x^2 plus 1/3! x^3,
plus and so on.
So all of these
numbers are 1, and all
you wind up with is the
factorials in the denominators.
That's the power series for e^x.
This was a discovery of Leonhard
Euler in 1740 or something.
Yes, Ma'am.
AUDIENCE: When you're
writing out the power series,
how far do you have
to write it out?
PROFESSOR: How far do you
have to write the power series
before it becomes well defined?
Before it's a satisfactory
solution to an exam problem,
I suppose, is another way
to phrase the question.
Until you can see
what the pattern is.
I can see what the pattern is.
Is there anyone who's
in doubt about what
the next term might be?
Some people would
tell you that you
have to write the
summation convention thing.
Don't believe them.
If you right out enough
terms to make it clear,
that's good enough.
OK?
Is that an answer for you?
AUDIENCE: Yes, Thank you.
PROFESSOR: OK, so
that's a basic example.
Let's do another basic
example of a power series.
Oh yes, and by the way, whenever
you write out a power series,
you should say what the
radius of convergence is.
And for now, I will
just to tell you
that the radius of convergence
of this power series
is infinity; that
is, this sum always
converges for any value of x.
I'll say a little more
about that in a few minutes.
Yeah?
AUDIENCE: So which functions
can be written as power series?
PROFESSOR: Which functions can
be written as power series?
That's an excellent question.
Any function that has
a reasonable expression
can be written as
a power series.
I'm not giving you a very good
answer because the true answer
is a little bit complicated.
But any of the
functions that occur
in calculus like sines,
cosines, tangents, they all have
power series expansions, OK?
We'll see more examples.
Let's do another example.
Here's another example.
I guess this was example one.
So, this example, I think,
was due to Newton, not Euler.
Let's find the power series
expansion of this function:
1/(1+x).
Well, I think that
somewhere along the line,
you learned about the geometric
series which tells you
that-- which tells you
what the answer to this is,
and I'll just write it out.
The geometric series tells
you that this function
can be written as an
alternating sum of powers of x.
You may wonder where
these minuses came from.
Well, if you really think
about the geometric series,
as you probably remembered,
there was a minus sign here,
and that gets replaced
by these minus signs.
I think maybe Jerison
talked about this also.
Anyway, here's
another basic example.
Remember what the
graph of this function
looks like when x = -1.
Then there's a
little problem here
because the
denominator becomes 0,
so the graph has a pole there.
It goes up to
infinity at x = -1,
and that's an indication that
the radius of convergence
is not infinity.
Because if you try to converge
to this infinite number
by putting in x = -1, here,
you'll have a big problem.
In fact, you see when
you put in x = -1,
you keep getting
1 in every term,
and it gets bigger and
bigger and does not converge.
In this example, the
radius of convergence is 1.
OK, so, let's do
a new example now.
Oh, and by the way,
I should say you
can calculate these numbers
using Taylor's formula.
If you haven't seen
it, check it out.
Calculate the iterated
derivatives of this function
and plug in x = 0 and see
that you get +1, -1, +1, -1,
and so on.
Yes sir.
AUDIENCE: For the
radius of convergence
I see that if you do
-1 it'll blow out.
If you put in 1 though, it
seems like it would be fine.
PROFESSOR: The
questions is I can
see that there's a
problem at x = -1,
why is there also
a problem at x = 1
where the graph is
perfectly smooth
and innocuous and finite.
That's another
excellent question.
The problem is that if you
go off to a radius of 1
in any direction and there's
a problem, that's it.
That's what the radius
of convergence is.
Here, what does happen
if I put an x = +1?
So, let's look at
the partial sums.
Do x = +1 in your mind here.
So I'll get a partial sum 1,
then 0, and then 1, and then 0,
and then 1.
So even though it doesn't
go up to infinity,
it still does not converge.
AUDIENCE: And
anything in between?
PROFESSOR: Any of
these other things
will also fail to
converge in this example.
Well, that's the only two
real numbers at the edge.
Right?
OK, let's do a
different example now.
How about a trig function?
The sine of x.
I'm going to compute the power
series expansion for sin(x).
and I'm going to do it
using Taylor's formula.
So Taylor's formula
says that I have
to start computing
derivatives of sin(x).
Sounds like it's going
to be a lot of work.
Let's see, the derivative
of the sine is the cosine.
And the derivative
of the cosine,
that's the second derivative
of the sine, is what?
Remember the minus,
it's -sin(x).
OK, now I want to take the third
derivative of the sine, which
is the derivative
of sine prime prime,
so it's the derivative of this.
And we just decided
the derivative of sine
is cosine, so I
get cosine, but I
have this minus sign in front.
And now I want to
differentiate again,
so the cosine
becomes a minus sine,
and that sign cancels with this
minus sign to give me sin(x).
You follow that?
It's a lot of -1's
canceling out there.
So, all of a sudden, I'm
right back where I started;
these two are the same and the
pattern will now repeat forever
and ever.
Higher and higher
derivatives of sines
are just plus or minus
sines and cosines.
Now Taylor's formula says I
should now substitute x = 0
into this and see what
happens, so let's do that.
When x is equals to 0, the
sine is 0 and the cosine is 1.
The sine is 0, so
minus 0 is also 0.
The cosine is 1, but
now there's a minus one,
and now I'm back
where I started,
and so the pattern will repeat.
OK, so the values
of the derivatives
are all zeros and
plus and minus ones
and they go through that
pattern, four-fold periodicity,
over and over again.
And so we can write
out what sin(x)
is using Taylor's formula,
using this formula.
So I put in the value
at 0 which is 0, then
I put in the derivative
which is 1, multiplied by x.
Then, I have the second
derivative divided by 2!,
but the second
derivative at 0 is 0.
So I'm going to
drop that term out.
Now I have the third
derivative which is -1.
And remember the 3!
in the denominator.
That's the coefficient of x^3.
What's the fourth derivative?
Well, here we are, it's
on the board, it's 0.
So I drop that term out
go up to the fifth term,
the fifth power of x.
Its derivative is now 1.
We've gone through the pattern,
we're back at +1 as the value
of the iterated derivative,
so now I get 1/5! x^5.
Now, you tell me, have we
done enough terms to see
what the pattern is?
I guess the next
term will be a -1/7!
x^7, and so on.
Let me write this out
again just so we have it.
x^3 / 3!-- So it's
x minus x^3 / 3!
plus x^5 / 5!.
You guessed it, and so on.
That's the power
series expansion
for the sine of x, OK?
And so, the sign alternate,
and these denominators
get very big, don't they?
Exponentials grow very fast.
Let me make a remark.
R is infinity here.
The radius of convergence
of this power series
again is infinity, and
let me just say why.
The reason is that the general
term is going to be like
x^(2n+1) / (2n+1)!.
An odd number I can
write as 2n + 1.
And what I want to
say is that the size
of this, what happens
to the size of this as n
goes to infinity?
So let's just think about this.
For a fixed x, let's
fix the number x.
Look at powers of x and
think about the size
of this expression when
n gets to be large.
So let's just do
that for a second.
So, x^(2n+1) / (2n+1)!, I
can write out like this.
It's x / 1 times x / 2
-- sorry -- times x / 3,
times x / (2n+1).
I've multiplied x by itself
2n+1 times in the numerator,
and I've multiplied
the numbers 1, 2, 3, 4,
and so on, by each other
in the denominator,
and that gives me the factorial.
So I've just written
this out like this.
Now x is fixed, so maybe
it's a million, OK?
It's big, but fixed.
What happens to these numbers?
Well at first,
they're pretty big.
This is 1,000,000 / 2,
this is 1,000,000 / 3.
But when n gets to be--
Maybe if n is 1,000,000,
then this is about 1/2.
If n is a billion, then this
is about 1/2,000, right?
The denominators keep
getting bigger and bigger,
but the numerators stay
the same; they're always x.
So when I take the product,
if I go far enough out,
I'm going to be multiplying,
by very, very small numbers
and more and more of them.
And so no matter what
x is, these numbers
will converge to 0.
They'll get smaller and
smaller as x gets to be bigger.
That's the sign that x is inside
of the radius of convergence.
This is the sign for
you that this series
converges for that value of x.
And because I could do
this for any x, this works.
This convergence to
0 for any fixed x.
That's what tells
you that you can
take-- that the radius of
convergence is infinity.
Because in the
formula, in the fact,
in this property that
the radius of convergence
talks about, if R is
equal to infinity,
this is no condition on x.
Every number is less than
infinity in absolute value.
So if this convergence
to 0 of the general term
works for every x, then radius
of convergence is infinity.
Well that was kind
of fast, but I
think that you've heard
something about that
earlier as well.
Anyway, so we've got the
sine function, a new function
with its own power series.
It's a way of computing sin(x).
If you take enough
terms you'll get
a good evaluation of sin(x).
for any x.
This tells you a lot
about the function sin(x)
but not everything at all.
For example, from
this formula, it's
very hard to see that the
sine of x is periodic.
It's not obvious at all.
Somewhere hidden away
in this expression
is the number pi, the
half of the period.
But that's not clear from
the power series at all.
So the power series are
very good for some things,
but they hide other
properties of functions.
Well, so I want to spend
a few minutes telling you
about what you can do
with a power series,
once you have one, to get new
power series, so new power
series from old.
And this is also called
operations on power series.
So what are the things that
we can do to a power series?
Well one of the things
you can do is multiply.
So, for example, what if
I want to compute a power
series for x sin(x)?
Well I have a power series
for sin(x), I just did it.
How about a power series for x?
Actually, I did that here too.
The function x is a
very simple polynomial.
It's a polynomial where
that's 0, a_1 is 1,
and all the other
coefficients are 0.
So x itself is a power
series, a very simple one.
sin(x) is a powers series.
And what I want to
encourage you to do
is treat power series
just like polynomials
and multiply them together.
We'll see other operations too.
So, to compute the power series
for x sin(x), of I just take
this one and multiply it by x.
So let's see if I
can do that right.
It distributes through:
x^2 minus x^4 / 3!
plus x^6 / 5!, and so on.
And again, the
radius of convergence
is going to be the smaller of
the two radii of convergence
here.
So it's R equals
infinity in this case.
OK, you can multiply
power series together.
It can be a pain if the
power series are very long,
but if one of them is
x, it's pretty simple.
OK, that's one thing I can do.
Notice something by the way.
You know that even
and odd functions?
So, sine is an odd function,
x is an odd function,
the product of two odd
functions is an even function.
And that's reflected in the fact
that all the powers that occur
in the power series are even.
For an odd function, like the
sine, all the powers that occur
are odd powers of x.
That's always true.
OK, we can multiply.
I can also differentiate.
So let's just do a
case of that, and use
the process of
differentiation to find out
what the power
series for cos(x) is
by writing the cos(x) as
the derivative of the sine
and differentiating
term by term.
So, I'll take this
expression for the power
series of the sine and
differentiate it term by term,
and I'll get the power
series for cosine.
So, let's see.
The derivative of x is one.
Now, the derivative of x^3 is
3x^2, and then there's a 3!
in the denominator.
And the derivative of x^5
5x^4, and there's a 5!
in the denominator,
and so on and so on.
And now some
cancellation happens.
So this is 1 minus, well, the
3 cancels with the last factor
in this 3 factorial
and leaves you with 2!.
And the 5 cancels with the
last factor in the 5 factorial
and leaves you with a 4!
in the denominator.
And so there you go, there's
the power series expansion
for the cosine.
It's got all even powers of x.
They alternate, and you have
factorials in the denominator.
And of course, you could
derive that expression
by using Taylor's formula, by
the same kind of calculation
you did here, taking higher
and higher derivatives
of the cosine.
You get the same
periodic pattern
of derivatives and values
of derivatives at x = 0.
But here's a cleaner way to
do it, simpler way to do it,
because we already knew
the derivative of the sine.
When you differentiate, you keep
the same radius of convergence.
OK, so we can
multiply, I can add too
and multiply by a
constant, things like that.
How about integrating?
That's what half of this
course was about isn't it?
So, let's integrate something.
So, the integration I'm
going to do is this one:
the integral from 0
to x of dt / (1+x).
What is that integral
as a function?
So, when I find the
anti-derivative of this,
I get ln(1+t), and then when
I evaluate that at t = x,
I get ln(1+x).
And when I evaluate the natural
log at 0, I get the ln 1,
which is 0, so this
is what you get, OK?
This is really valid, by the
way, for x bigger than -1.
But you don't want to think
about this quite like this
when x is smaller than that.
Now, I'm going to try to apply
power series methods here
and find-- use this integral
to find a power series
for the natural log, and I'll
do it by plugging into this
expression what the power
series for 1/(1+t) was.
And I know what that is because
I wrote it down on the board
up here.
Change the variable
from x to t there,
and so 1/(1+t) is 1 minus t
plus t^2 minus t^3, and so on.
So that's the thing in the
inside of the integral,
and now it's legal to
integrate that term by term,
so let's do that.
I'm going to get something
which I will then
evaluate at x and at 0.
So, when I integrate 1 I get
x, and when I integrate t,
I get t.
I'm sorry.
When I integrate t, I get t^2
/ 2, and t^2 gives me t^3 / 3,
and so on and so on.
And then, when I
put in t = x, well,
I just replace all the t's by
x's, and when I put in t = 0,
I get 0.
So this equals x.
So, I've discovered that ln(1+x)
is x minus x^2 / 2 plus x^3 / 3
minus x^4 / 4, and
so on and so on.
There's the power series
expansion for ln(1+x).
And because I began
with a power series
whose radius of
convergence was just 1,
I began with this power
series, the radius
of convergence of this
is also going to be 1.
Also, because this function,
as I just pointed out,
this function goes bad when
x becomes less than -1,
so some problem happens,
and that's reflected
in the radius of convergence.
Cool.
So, you can integrate.
That is the correct power series
expansion for the ln(1+x),
and another victory of Euler's
was to use this kind of power
series expansion to calculate
natural logarithms in a much
more efficient way than
people had done before.
OK, one more property, I think.
What are we at here, 3?
4.
Substitute.
Very appropriate for me
as a substitute teacher
to tell you about substitution.
So I'm going to try to find
the power series expansion
of e^(-t^2).
OK?
And the way I'll do that is
by taking the power series
expansion for e^x,
which we have up there,
and make the substitution x =
-t^2 in the expansion for e^x.
Did you have a question?
AUDIENCE: Well,
it's just concerning
the radius of convergence.
You can't define x so that is
always positive, and if so,
it wouldn't have a radius
of convergence, right?
PROFESSOR: Like I say, again the
worry is this ln(1+x) function
is perfectly well
behaved for large x.
Why does the power series
fail to converge for large x?
Well suppose that
x is bigger than 1,
then here you get
bigger and bigger powers
of x, which will
grow to infinity,
and they grow large faster
than the numbers 2, 3, 4, 5, 6.
They grow exponentially, and
these just grow linearly.
So, again, the general term,
when x is bigger than one,
the general term will
go off to infinity,
even though the function
that you're talking about,
log of net of 1 plus
x is perfectly good.
So the power series is not
good outside of the radius
of convergence.
It's just a fact of life.
Yes?
AUDIENCE: [INAUDIBLE]
PROFESSOR: I'd rather--
talk to me after class.
The question is why is
it the smaller of the two
radii of convergence?
The basic answer
is, well, you can't
expect it to be bigger than that
smaller one, because the power
series only gives
you information
inside of that range
about the function, so.
AUDIENCE: [INAUDIBLE]
PROFESSOR: Well, in this
case, both of the radii
of convergence are infinity.
x has radius of convergence
infinity for sure,
and sin(x) does too.
So you get infinity
in that case, OK?
OK, let's just do
this, and then I'm
going to integrate
this and that'll
be the end of what I
have time for today.
So what's the power
series expansion for this?
The power series
expansion of this
is going to be a
function of t, right,
because the variable here is t.
I get it by taking my expansion
for e^x and putting in what x
is in terms of t.
Whoops!
And so on and so on.
I just put in -t^2 in place of
x there in the series expansion
for e^x.
I can work this out
a little bit better.
-t^2 is what it is.
This is going to give me a t^4
and the minus squared is going
to give me a plus,
so I get t^4 / 2!.
Then I get (-t)^3, so there'll
be a minus sign and a t^6
and the denominator 3!.
So the signs are
going to alternate,
the powers are all even,
and the denominators
are these factorials.
Several times as this
course has gone on,
the error function has
made an appearance.
The error function was, I guess
it gets normalized by putting
a 2 over the square
root of pi in front,
and it's the integral of
e^(-t^2) dt from 0 to x.
And this normalization
is here because as x
gets to be large
the value becomes 1.
So this error function is
very important in the theory
of probability.
And I think you calculated
this fact at some point
in the course.
So the standard definition of
the error function, you put a 2
over the square
root of pi in front.
Let's calculate its
power series expansion.
So there's a 2 over
the square root of pi
that hurts nobody
here in the front.
And now I want to
integrate e^(-t^2),
and I'm going to use this
power series expansion for that
to see what you get.
So I'm just going to
write this out I think.
I did it out carefully in
another example over there,
so I'll do it a
little quicker now.
Integrate this term
by term, you're
just integrating powers of
t so it's pretty simple,
so I get-- and then I'm
evaluating at x and then at 0.
So I get x minus x^3 /
3, plus x^5 / (5*2!),
5 from integrating
the t^4, and the 2!
from this denominator
that we already had.
And then there's a -x^7
/ (7*3!), and plus,
and so on, and you can imagine
how they go on from there.
I guess to get this
exactly in the form
that we began talking about,
I should multiply through.
So the coefficient of x is 2
over the square root of pi,
and the coefficient of x^3 is
-2 over 3 times the square root
of pi, and so on.
But this is a perfectly good
way to write this power series
expansion as well.
And, this is a very good way to
compute the value of the error
function.
It's a new function
in our experience.
Your calculator
probably calculates it,
and your calculator probably
does it by this method.
OK, so that's my sermon
on examples of things
you can do with power series.
So, we're going to do the
CEG thing in just a minute.
Professor Jerison wanted
me to make an ad for 18.02.
Just in case you were thinking
of not taking it next term,
you really should take it.
It will put a lot of
things in this course
into context, for one thing.
It's about vector
calculus and so on.
So you'll learn about
vectors and things like that.
But it comes back and
explains some things
in this course that might
have been a little bit
strange, like these strange
formulas for the product
rule and the quotient rule and
the sort of random formulas.
Well, one of the things
you learn in 18.02
is that they're all special
cases of the chain rule.
And just to drive
that point home,
he wanted me to show you
this poem of his that
really drives the points
home forcefully, I think.
