PROFESSOR: Hello, hello
again, welcome back.
Today we're finally going
to talk about power series.
Now, we've been
hinting at power series
and using that word in
the last several videos.
Because we're
really switching now
from talking about
series in terms of,
does this converge or
diverge and running
through all those various
tests that we've had.
Now, in the last
couple of lessons here,
we're going to switch to really
using them for something,
right?
And maybe, you know,
it crossed your mind.
It would be amazing
if it hadn't.
What is all this stuff good for?
So the unfortunate
truth is, you don't
get to see the really,
really cool stuff this
is good for until
you get further
on into other classes, just
the nature of the beast, right?
But what we're doing with
series is a big deal.
And it's a major--
because like I said
before, that it's the way
that multiplication is
necessary to learn before you
can go and talk about algebra.
This stuff is necessary to learn
before you can go on and do
stuff in engineering and
physics and especially
in higher math classes.
Power series are a
major, major tool which
we'll use in other places.
And so the really cool
things we can do with them
will be in other classes.
So we're just sort of giving
you the foundation right now.
But all this stuff
with does this converge
and is that converge, is
really to lay the groundwork
for power series.
So let's get started.
What's a power series?
So here's our
definition, which may not
look terribly
enlightening either
when you first look at it.
But it's worth pointing out
a couple of the pieces here.
So first off, we
have a sum, right?
We've seen those before
when we do a series.
We have an ak.
So that's some sequence of
numbers, some list, right,
the ak's.
But then what makes this a power
series is this extra piece, x
minus c to the k.
and x now is a variable.
So this x here is a variable.
This c is some number, right?
And we'll say, we have a
power series centered at c.
Where that centered means,
we'll explain as we go along.
Right now, I don't see a circle
anywhere happening, right?
But we will.
But this c is where
we're centered.
And it's just a number, right?
So this makes more
sense, I think,
if we start writing out
the terms in this series.
So for example, when
k is equal to zero,
we would have a zero times
x minus c to the zero power.
Well, that would just be one.
So we just get a zero.
Then when k is 1,
we get a1 times x
minus c to the first power.
You hear me saying, power.
That's what the word
power means, right?
It's x minus c raised
to various powers.
So when we write out the
terms long ways-- and I really
encourage you to
take a little while
and look at what
I've written here.
And let this sink into your head
as what a power series really
is.
It looks, well, it looks like
a really big, long polynomial,
right, a0 then
another number, a1,
times x minus c,
another number, a2,
x minus c squared and so on.
So the x minus c
raised to powers
is where this gets its name.
Now, this is even
more obvious maybe as
to what this is supposed to look
like and how does this it act,
if we think about what
happens if c is equal to zero.
So if the c were zero, then you
wouldn't see it in the formula,
right? x minus c
would just be x.
So what you'd get is
something that oh, well, it's
written like this,
that obviously
looks like a polynomial, right?
Number plus number times x
plus number times x squared,
and so on.
So here's the key
thing to take away.
Just like we've seen before,
that a sequence is a list.
And a series is a list added up.
A power series is
a kind of function.
Whether or not it
converges and what
it converges to depends on
the x you plug in there.
x is a variable, right?
So that's really cool.
As we change the
x's, then the series
may or may not even
converge anymore.
And what it converges to,
well, that's the value
of the function, when
we plug in the x, right?
What you should think
of in your head when
you see a power
series is, they're
like infinitely
long polynomials,
like polynomials that go
on forever and forever.
And that's way more accurate,
right, way more to the point
than even the pictures
we've drawn on the screen
here looks like, right?
These things really do act
a lot like polynomials.
But because they're
infinitely long,
they can do more powerful,
in the sense of super powers,
and more awesome stuff that
a polynomial ever could.
Let me give you an example.
We've already sort of tiptoed
around this one a little bit.
So this is the series k equal
zero to infinity of x to the k.
It's a power series, right?
It's centered around zero.
Because you see just x,
right, not x minus 7,
where it would be
centered around 7.
It's x by itself.
So really you can think
of that as x minus zero
all raised to the k power.
So its centered at zero.
The coefficients,
those ak's, are all 1.
That's why we don't notice them.
They are all just 1 times
x minus zero to the k.
So it's definitely
a power series,
x raised to all
these various powers.
But if you look at
it closer, right, you
see that we've seen
stuff like this
before, if we use a
different letter, right?
We used to talk about
geometric series.
And we would have
like r to the k.
And maybe there was an
a in front, a constant.
In this case, that would be 1.
If you plug in a number
for x, what we get
is a geometric series, right?
We're using x now instead
of r for obvious reasons.
Because we're going
to use x's whenever
we talk about power series.
But this is just like
a geometric series,
where we think of the
r as a variable, right?
So of course, it's going
to converge whenever
the absolute value of the common
ratio-- in this case, it's x.
Whenever the up to value
of x is less than 1.
OK, and even better than that,
right, I think this is worth,
again, sayings that
you keep this in mind.
Power series is a
function, right?
Again, sequence is a list.
Series is a list added up.
A power series is a
function that you get
by raising x minus c to powers.
So what function is this?
Well, whenever you plug in x,
you get a geometric series.
So when it converges, it
converge to the first term.
Which is this case, will
be 1 over 1 minus x.
In other words, what we've got
here is a weird, freaky way,
it looks like, to write the
function 1 over 1 minus x.
Then of course, doesn't
always converge, right,
if I plug in a 15, then it's
not going to converge anymore.
If I plug in a number
between negative 1 and 1,
it will converge, right?
And so you might very well be
asking, why do I care, right?
1 over 1 minus x,
that's a great function.
Well, this weird series thing
looks bizarre and strange.
Who would want
something like that?
The answer is, you do.
You just don't know it yet.
This is really, really important
and really useful and really
cool stuff.
And again, the why and
the how for using these,
the coolest examples show
up in later classes, right?
It's sort of like, you
know, third grader going,
why do I have to
know how to multiply?
Adding was fine, right?
Anyway, once you
know how to multiply,
then you can do all sorts of
really cool stuff with it.
And you know, where you
would even make sense
to ask the questions, you
know, about the things
that you can actually solve with
multiplication until you have
learned it really thoroughly.
And that's almost the
situation we're in now.
Where you're getting tools and
how you use them, you know,
is unfortunately,
because we're almost out
of time in this course,
left for other classes.
Let's look at the
kind of example
that you'll find the
homework full of, right?
This is the standard
kind of question
that you see in the homework for
an introductory lesson on power
series.
So we have some sort
of a power series.
And the question is, find all
the x's where this converges.
Or another way to
ask this question is.
Here's a power series.
That means it's a
function, right?
It's a function of x.
What's the domain
of this function?
That's the other way of
asking the same question.
What numbers can I plug in
there for x and it makes sense?
So well, OK, first off,
notice this is a power series.
It's centered at 5, because
we have x minus 5 to the k.
And the ak, the coefficients,
we call them, right?
Those are these numbers that
go in front of the x minus 5,
raised to various powers.
So the ak's in this
example, all right,
those coefficients are 3
over 2 to the k times k
and the denominator there, OK?
So what are the values of x
where this thing will converge?
Well, we've-- because figuring
out what power series are is
kind of a jolt to
your system, right?
We had sequences.
And we let that sink
in for a little bit.
And we had series.
And we were taking a long
time to let that sink in.
And power series,
being a function,
is a really different
kind of animal.
And that jolt to your system is
kind of enough for one video.
So we've hinted at this in
several other videos, right?
Where we've had
problems like this.
And the trick was to use
the ratio test or maybe
the root test and try to
figure out where it converges.
In particular, what
we're going to do
is what we call the
generalized ratio test.
Where we throw absolute values
around the terms in the series
and see if it converges
absolutely, OK?
So what I'm going to do is put
absolute values around this
and do the ratio test.
OK, so that was a lot of
formulas just popped up there
for clicking one button.
So let's look and
see what we've got.
If you check, you'll see that
this first piece is the k
plus 1 term in our series.
Notice that this
is x minus 5 raised
to the k plus one power 2.
Then the 2 to the k becomes
a 2 to the k plus 1.
The k in the denominator
gets a k plus 1.
Then we need to divide
by the terms as written,
in other words, flip
it, right, and multiply.
Which gives me 2 to the k and k
over 3 and x minus 5 to the k.
Standard way we
do the ratio test.
But again, I'm putting
absolute values around it,
because I'm not just testing
to see whether it converges.
I'm testing to see whether
it converges absolutely.
Now, the cool thing
about the ratio test,
whenever you do is
with a power series,
is look what happens
to my x minus 5.
The x minus 5 to the
k in the denominator
will cancel with all but one of
the x minus 5 to the k plus 1
in the numerator.
So I'll be left with
just a x minus 5.
Of course, the 3's cancel,
the 2 to the k's cancel
and leave me just one extra 2.
And I'll have a k and
a k plus one leftover.
So this is all the
leftover pieces.
Also, notice that
k's always positive.
It's going off to infinity.
2 is positive. k
plus one is positive.
The only thing that might
be negative in this whole
situation is the x minus 5.
Because I don't know that
x isn't 4, for example.
So I'm going to leave
the absolute values just
around the x minus
5, because everything
else in that fraction
is already positive.
Well, now, I take the limit
is k is off to infinity
and what happens?
Well, the k in the
top of the bottom tile
in the tower of power
is one way to say that.
So we end up with a 2
k in the denominator
and an absolute value of
x minus 5 k in the top.
Remember, it's k
that's going off
to infinity, the x isn't, right?
The x is just x.
So it's just like having
another number here.
So when we look at the
tower of power, we get well,
the number multiplied
by the k in the top
over the number multiplied
by the k on the bottom.
Because of the way the tower
of power works they tie.
Again, one thing that sometimes
confuses people is they
think, oh, I'm letting
k go off to infinity.
And they want to let k and x
and everything go to infinity.
But it's just the k's that
are going to infinity.
x is whatever x is, right?
It's not moving.
So that leaves us with absolute
value of x minus 5 over 2.
OK, great well, what
does that tell us?
Well, whenever you do the
ratio test, you do the limit,
see what you get at the end.
If the thing at the
end is less than 1,
then we know we
converge, in this case,
converging absolutely.
Because I started out by
putting absolute values there.
And if what we get at the end is
bigger than 1, then of course,
we'll diverge.
And if its equal to 1, we don't.
We'll have to deal
with it separately.
And we'll come back to that
thought in just a second.
So the deal is we will
converge, in fact, absolutely.
Because I put absolute
values around the things
when I started,
whenever absolute value
of x minus 5 over
2 is less than 1.
Or better yet, let me take
the 2 to the other side.
And [INAUDIBLE],, we get
absolute value of x minus 5
is less than 2.
And that leaves us
a really simple,
you know, eighth grade math
level of absolute value,
inequality to solve.
So it's super easy
to solve these.
But I want to take a minute
to talk about how you do it
and why the answer
is what it is.
Because I think it will make
the reason some of the stuff
we're going to talk
about later works,
if you really understand what
we're about to say right now.
So maybe the way you would have
seen solving absolute values
and inequalities before
would have been to say,
the opposite value of x minus
5 less than 2 means, x minus 5
is less than positive
2 or x minus 5
is bigger than
negative 2, right?
And then you proceed from thee.
That's what we usually
tell eighth graders
when they solve this for
the very first time, right,
with absolute values.
It's really not necessarily
the best way to do it
or to think about it.
And so I'm going to-- maybe
you've never even seen this
and so it's work explaining
why that even actually works.
Because if you want to
work the problems this way,
it's perfectly OK.
Even though we will think
about it differently
in about a minute from now.
If the x minus 5 is
positive-- and we
don't know what
the x is, right. x
could be 7 or 12 or something.
In which case, x minus
5 would be positive.
So the absolute value
doesn't do anything.
So I could just get rid of it.
And I'll have x minus
5 is less than 2.
Of course, I could take
the 5 to the other side
and get x is less than 7, right?
On the other hand, x might
be a number smaller than 5.
It might be 4 or
3 or negative 200.
In which case, x minus
5 would be negative.
Well, then what happens
to the absolute value?
The absolute value
of a negative number
was supposed to take it and
turn it positive, right?
Well, how do you turn a
negative number positive?
How would you write that down?
I mean, I could write,
take this negative number
and make it positive, but
that's a lot of words.
What symbol tells you to
take that negative number
and make it positive?
What symbol tells
you to switch sides?
Well, you put a minus
sign in front of it.
Now, this may look
really, really weird.
But remember, if x were
say, 4 or 3 or something,
so that x minus 5 were
a negative number, then
putting a minus sign in front
of that is what flips a sign
and makes it positive.
So the absolute
value of x minus 5,
even though it
looks really weird,
would be negative x
minus 5 if the x minus 5
itself were already negative.
Because that extra minus sign
is what would flip the sign
to make it positive.
So the other case is negative
x minus 5 is less than 2.
Now, what do I do
with the minus sign?
Well, remember I could
multiply through the minus sign
and get rid of it or move
it to the other side.
But that switches the
direction of the inequality.
So we get x minus 5 is
bigger than negative 2.
And again, you were probably
told this in high school
somewhere when you were
solving inequalities
with absolute values,
that you just do x minus 5
is less than 2.
And x minus 5 is
bigger than negative 2.
And nobody probably
ever explained why.
But this is why.
Now, we take the 5
to the other side
and we get x is
bigger than negative 2
plus 5, which of course, is 3.
So we now have that x has
to be between 3 and 7.
OK, that's a perfectly
reasonable way to do that.
But I want to pause
for just a second and--
again, continue
our pausing, right?
Our discussion of how we
solve this inequality.
And I want to try doing
this with geometry
instead of algebra.
The reason is, if
you can understand
what I'm about to
work here, instead
of just doing the
algebra, it makes
the rule we're about to
have make a lot more sense.
And so let me just
throw this out, right.
If I were doing this in class,
I would talk to the class.
And everybody answer back.
With no one at the
other end of this,
they can't answer back to me.
Because I'm recording
this for future posterity.
It's a little weirder.
But what would happen
if I said, how far apart
are the numbers 9 and 6?
How far apart are 9 and 6?
And your brain immediately
goes there are three, right?
But let us stop for a second.
What did your brain
just do though?
How did you get three?
Well, you said 9 minus the
6 gives us three, right?
So how far apart are 13 and 15?
And again, your brain jumped
in and just said, 13 and 15,
they're two apart.
And how you get that?
Well, if you ask your brain,
let it think for a second,
you'd realize that what you
did was 15 minus 13, right?
In other words, it was the big
number minus the small number.
Now, of course, that should
sound really familiar.
Because we did that a
lot at the very beginning
of the semester when we were
doing volumes and things.
Where we were taking big
number minus small number
to find the distance
all the time.
We were pumping
water out of tanks
and all kinds of word problems.
And it was always the big
number minus the small number
to find the distance,
that things had to move
or how far apart they
were or whatever.
Well, how far apart is x and 5?
Oh, well, I don't
know what x is.
So I don't know whether
it ought to be x minus 5.
Or whether it ought
to be 5 minus x.
Because I don't know
which one's the biggest.
Does that make sense?
So how do we fix that?
Well, what we do is
we just subtract them
in any order, right?
I prefer x minus 5, because
that's the one on the screen
right now.
And I might get an
answer that's negative.
So to fix it, I just throw
absolute values around, right?
So what is the absolute
value of x minus 5 really?
The geometry answer is, it's
just the distance from x to 5.
Again, we don't know whether
it should be x minus 5 or 5
minus x.
But if I put absolute
values around it,
it doesn't matter anymore.
So I'll always have the
distance from x to 5,
if I look at the absolute
value of x minus 5.
What that means is, I could
draw myself a 5 and then
read that inequality, absolute
value of x minus 5 less than 2.
I could read that instead and
say, the distance from x to 5
is less than 2.
And then I ask myself,
where can x live, right?
If the distance from x to 5
has to be less than 2, means
I could stand at 5 and go two
units in either direction,
so all the way up to 7
or all the way down to 3.
And anywhere in that range,
right, two on either side,
will work, right?
x could be 3 and 1/2.
It could be 6.9.
Anywhere in that
range, I'm good.
The distance from x to that
number or 5 that number
will be less than 2.
And I'm not counting
the 3 or the 7,
because that distance is
actually equal to 2, OK.
So this is a big deal, right?
Because this is going
to show up over and over
and over in our discussions.
In fact, we're going have a
name for this kind of thing,
this idea of standing at the 5.
And remember, what was the 5?
5 was the number that our power
series was centered around.
And now, we get to see where
that language comes from.
It was a power
series centered at 5.
And why is it centered?
Because it converges 2
on either side, right?
That's the language
of centered and circle
and shows up here,
because, of course,
it's the same on either side.
In fact, the language of circle
extends a little further.
We'll call that 2, the
radius of convergence.
So if I stand at 5 and
I go 2 on either side,
that gives me the interval of
convergence from 3 to 7, right?
Now, I have to worry
about whether I
should count the 3 or the 7.
Because that will
take a little effort.
But what you typically
see in the homework
or on the exams is,
here's a power series,
find its radius of convergence.
In this case, it's
going to be 2, right?
And we can get that by looking
at the inequality we got.
And then what's its
interval of convergence?
For the interval of
convergence, we'll
have to be a little careful.
Because we still have
to check the endpoints.
Because when x is 3 and when
x is 7, well, in that case,
the absolute value of
x minus 3 would be--
sorry, the absolute value of
x minus 5, if x is 3 or 7,
would be equal to 2.
In which case, we would
get 1 from several minutes
ago when we did the ratio test.
In other words, if
x were 3 or x were 7
and we did the ratio
test, we get 1, right?
If x is bigger than
7 or less than 3
and you did the ratio
test and its bigger than 1
and would definitely diverged.
But at 3 and at 7 you
get 1 from the ratio test
and anything could
have happened.
So we have to check
endpoints separately.
So we just do it.
Let's plug in 7 and
see what we get.
So I would have the series
with a 3 over a 2 to the k
and a k, then suddenly
7 minus 5 to the k.
And now, I think you see why
7 was an important number.
Because 7 minus 5 to
the k is 2 to the k.
And so obviously, that 2 to the
k is going to cancel with the 2
to the k in the denominator.
That's when our series quit
looking geometrically, right,
with a number to
the k and started
looking like harmonic
series, which is exactly
harmonic series times 3.
And we know that
guy diverges, OK.
Now, sometimes it
confuses people.
Because they think,
when I get to this step
and I have a series
there, what do I do?
And the answer
is, well, you just
go to war with it
using whatever tools
from the last several
videos that you have to do,
maybe alternating series
test or divergence test.
Or it may it just be a series
we know, like in this example.
Anything is fair game.
And whatever happens,
you just have
to use all those
things that we learned
before to figure it out.
So if x is equal to 3,
did I get a 3 minus 5
raised to the k-th power?
So that's a little different
than the one we had before.
Because 3 minus 5 is negative 2.
So what will happen is,
I'll take that negative 2
and write it as negative
1 and positive 2.
And then both of those get
raised to the k-th power.
So in other words,
what I have is a 2
to the k, which will
cancel with the 2
to the k in the denominator
and a negative 1 to the k
that's left over.
So what's left with the
negative 1 in the casing
there is an alternating
harmonic series.
And of course, if we
remember from our videos
on alternating series, this
guy will definitely converge.
So our radius of
convergence will be 2.
The interval of convergence
will go from 3 to 7.
But here we include the 3.
And we don't include the 7.
And if it's been
a while since you
though about interval notation--
and we'll use interval notation
quite a lot, especially
in the homework
will ask for answers
in interval notation.
The parentheses means,
we don't include the 7.
And the little bracket
means, we do include the 3.
How do I remember that?
What I always tell people when
I'm teaching college algebra
or something and people see
this for the first time,
I tell them that the bracket
almost looks like a capital
letter I. Like a
capital letter I
when our printer is running
low on ink and half of it
didn't print?
But cal is saying, capital
letter I for include the 3.
On the other hand, the 7
is kind of bending, right?
The parentheses is bending.
Well, ew, I don't want
to touch that 7, right?
So however it works for
you to remember that.
The interval of convergence
will go from 3 to 7.
We're including the 3,
not including the 7.
So here's another example.
Here's a power
series centered at 2
and our coefficients in front.
That would be the k
raised to the k power.
So because we've got things
raised to the k-th power,
we could do the ratio test.
But let's try the root test this
time just A, to be different
and to show you that
both ways will work.
And B, because the root test
is really easy for this one.
So I'm going to use, well,
really the generalized root
test.
Because I'm going throw
absolute values around it first.
So remember the real
root test required
that all the terms be positive.
The generalized version means,
we put up some [INAUDIBLE]
around it and check to see
where it converges absolutely.
So if absolute
value is around here
and then the algebra
is really easy, right?
When I take the
k-th root, I just
get k, which is always positive.
So I don't need absolute
values around it anymore
and x minus 2, which you
know, might be negative.
So I'll leave the
absolute values there.
Now, when k goes
off to infinity,
oh, boy, what's
going to happen now?
k goes to infinity, the whole
thing goes to infinity, right?
So my ratio test would
tell me that, everything
is going to diverge and OK.
So we're done, well, almost.
The limit will be infinity,
unless the x were 2.
If x were 2, then
I would have zero
times k, which is always zero.
And the limit will be zero,
right, which is less than 1.
Which means, the ratio test
tell me we converge there.
So the series converges
only when x is 2.
And the radius of convergence,
in this case, I'll say,
it's zero.
Because I can stand at 2.
And I converge there.
And if I move in either
direction at all,
I don't converge anymore.
So I can move zero distance on
either side and still converge.
So the radius of
convergence is zero.
And I just converge at 2.
And of course, it's
pretty easy to see,
if you stop and think about it.
That we'll always
converge at the point
that we're centered around.
Because that x minus 2
raised to the k-th power,
if I plug in a 2, right, or if
I have a power series centered
at 17 and I plug in 17, I'll
get a whole bunch of zeros.
And I can add up infinitely
many numbers in my head,
as long as, you know,
all but one of them are.
So this was what
will happen whenever
we plug in the number for x
that we're centered around.
We'll always converge there.
But this is the great example
of the fact, that maybe
that it's the only place
where it converges.
OK, here's another one.
So I've written
this as a fraction.
Because we'll see it
this way later on.
But this is a power series.
It's centered at zero.
That's why we see
the x by itself.
So you could think of this
x minus zero all to the k.
The coefficients would
be 1 over k factorial.
And again, I've
taken that x and k
and written it in the numerator.
That's just a little
bit of algebra.
Probably because we'll
see it that way later.
Well, let's do the ratio test.
So I need to take
x to the k plus 1
divided by k plus 1 factorial.
Then we'll flip it into the
k factorial over x to the k.
And the factorial, of course,
is just screaming out,
ratio test me, right?
And it works, right?
Everything will cancel with the
k plus 1 and the k factorial
except for our 1 k plus 1
that's in the denominator.
And the x's, of
course, cancel, except
for one left over x, which
will be in the numerator.
The absolute value is not
necessary around the k plus 1.
That's definitely always
going to be positive.
But it is around the
x, because I don't
know what number x is yet.
When I want to take the
limit, what happens, right?
There's only a k
in the denominator.
So as k goes off to
infinity, the denominator
gets bigger, which
means, the whole thing
goes to zero no
matter what the x is.
So this series will converge,
no matter what the x is.
And again, remember,
I'm taken the limit
as k goes to infinity.
So x is a particular number
like 1,500 or 7 or negative 32.
x is just a number.
And it's not moving.
Then I'll let the k
go off to infinity.
And whatever number I
have that's not moving
divided by k plus 1, right?
And then I'll let the k
plus 1 go off to infinity,
don't get zero.
And whenever you get
zero from the ratio test,
it means that your
series converges.
So this series converges,
in fact, absolutely,
because I put absolute
values around things.
This series converges absolutely
no matter what the x is.
So what's my radius
of convergence?
Well, it's infinity.
And the interval will
go from minus infinity
to plus infinity.
OK, so you may think, wow,
we've done a lot of example
here, right, and
what's the point?
Well, the reason we did
these three examples
is, because we
have this theorem.
Whenever you have
a power series,
there are only three
possibilities, right?
And exactly one of these three
possibilities has to happen.
So you have a power series.
And this example is a
generic looking power series
centered at c.
Well, if I were to plug
in c that I have for x,
I'd have c minus
c, which would be
zero raised to the k-th power.
And I would be multiplying
by a whole bunch of zeros
and adding up a
whole bunch of zeros.
That means, that's
definitely going to converge.
So one possibility is it's
like the second example we did.
It could converge when x is c.
And maybe that's the
only place it converges.
The radius of convergence
might be zero.
Another possibility is what
happened in the last example.
It converged everywhere, doesn't
matter what you plug in for x.
It could converge.
What happens, I wouldn't
say, more often,
but more interestingly, right--
it means it's more work for you.
Is there some number r that we
call the radius of convergence,
right?
And it's positive, right?
It's not infinity.
So like the first example,
where the radius of convergence
was 2.
And our power series will
converge absolutely, right,
if we're within the radius
of convergence of the place
where it's centered.
In other words, if we're in
the interval of convergence,
right, so kind of the endpoint.
And we'll diverge
outside of that.
And again, we have to check
the endpoint separately.
At the endpoints, we
might converge absolutely
or it might converge
conditionally.
It's not always clear which way.
It could work, any of
those could happen.
So what this let's us do then
is ask really cool questions
like this one.
Suppose we have this power
series, ak x minus 3 to the k?
And some magical elf
shows up and tells you,
that it converges when
you plug in 8 for the x.
So it converges at x equals 8.
Then my question
is does it converge
when you plug in 6 for x?
Oh, that's an interesting
question, right?
I mean, I don't know
what the ak's are.
I can't go in and try to
do some sort of a test.
So how do I know
what's going on?
Well, let's draw a picture here.
So this is centered
around 3 remember.
And I know that I
converge when I plug in 8.
That means, that my
radius of convergence
has to be at least 5, right?
The interval of
convergence has to run,
you know, from 3
all the way up to 8.
So anywhere in there
is going to converge.
So I will definitely
converge when x is 6.
Do I converge at negative 1?
Well, let's see.
So I know that my
interval of convergence
runs all the way from
3 at least up to 8.
And notice my radius
is at least 5.
So if I start at 3 and go
5 in the other direction,
I would go down to negative 2.
So sure, I definitely got
to converge at negative 1.
Because it's inside that
radius of convergence.
I know the radius of convergence
has to be at least 5.
Well, do I converge
at negative 2?
Well, I converged at 8.
Well, the answer
is maybe, right?
If all I know is
that I converge at 8,
then all I really know is
the radius of convergence
is at least five.
I don't actually know
the radius of convergence
is 5, just that it's
at least 5, right?
It could have
converged other places.
So what happens at negative 2?
Well, the radius of convergence
might have been exactly 5.
And I might have
converged at one endpoint
and not at the other.
So maybe, I don't know whether
it converges at negative 2
or not.
Does it converge at 9?
Well, again, I don't
know whether it converges
at 9 or not, because
the problem just said,
that it converges at 8, right?
It didn't tell me what
happened pass that.
So I know that between 3
and 8, I've got to converge.
But just telling me,
you converged at 8,
it might have
converged everywhere.
I just don't know.
So this is a good place
maybe to stop and make
one quick comment
about language.
So I mark these
on people's exams
when we talked about sequences.
Obviously, I wasn't
going to count of for it.
But it's a useful place to stop
and notice the prepositions
that I'm using here.
So I will say that,
this series converges
at some point, converges at
8 or at negative 2 or at 9.
And the reason is, I'm
thinking of x as being
a place where I could be.
And I plug that number
into the power series
and see whether it
converges or not.
And so at is there, because I'm
thinking of this like a place,
right?
If I'm standing at 8 and I plug
that 8 into the power series,
then I would converge there, OK?
Sometimes people aren't
used to language, which
is perfectly understandable.
This is the first time
you seen this stuff.
And they'll talk
about a sequence.
What they really mean is a
sequence converges to 13 say.
And they'll write a
sequence converges at 13.
And that's not quite
the right language.
Because we'll usually
say, at, for this kind
of a problem, where we're
talking about a power series.
And at means, I could plug that
number into the power series
and it converge there.
I've got a sequence that
converges to something,
I'll use the preposition to.
And say, here's a sequence
that converges to something.
Again, it's not something
that you count for,
but it is something
that makes you
sound like you know what you're
talking about in later classes,
if you say it wrong.
So this is a good
place to practice
getting the prepositions
and the language right.
