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AP Calc: LIM‑8 (EU), LIM‑8.D (LO), LIM‑8.D.1 (EK), LIM‑8.D.2 (EK), LIM‑8.D.3 (EK), LIM‑8.D.4 (EK)

We've already seen many
examples of infinite series. But what's exciting about what
we're about to do in this video is we're going to use infinite
series to define a function. And the most common
one that you will see in your mathematical
careers is the power series. And I'm about to write a general
case of the power series. So I could imagine
a function, f of x, being defined as
the infinite sum. So going from n equals 0
to infinity of a sub n-- so a sub n is just going to be
the coefficient on each term-- times our variable x
minus some constant c. You could almost
imagine this is shifting our function to the n-th power. So if I were to
expand this out, I have my first term's
coefficient, a sub 0, times x minus c
to the 0-th power, plus a sub 1 times x minus
c to the first power. This one, of course, will
simplify just a sub 0. This would simplify to a sub
1 times x minus c plus a sub 2 times x minus c squared. And I could just keep
going on and on and on. Now, when you see
this, you might say, aren't our geometric
series, don't those look like a special
case of a power series if our common ratio was an x
instead of an r in that case, or if our common ratio was a
variable, I guess I could say? And you would be right. That absolutely
would be the case. So a geometric series. So let's just think about
defining a function in terms of a geometric series. And of course, we
don't have to use x all the time as the
independent variable, but this is kind of the
most typical convention. I guess we could also use r
as an independent variable if we wanted as well. But let's imagine
a function g of x. We could have g of r
if we wanted, but let's do g of x is equal
to the sum from n equals 0 to infinity
of a times x to the n. So this is kind of a typical
geometric series here. And what's the difference
between this and this? Well, the difference is is
here, for every term we're going to have the
same coefficient a, while over here we have a sub n. We're multiplying by a different
thing every time up here. We're multiplying by the
same thing over here. And in this case, this
particular geometric series I just made, instead of
having x minus c to the n, we have just x to the n. So you could say, well,
this is a special case when c is equal to 0. And we can expand it out. This is a times
x to the 0, which is just going to be a, plus
a times x to the first, plus a times x squared. And we just go on and
on and on forever. Now, what's exciting
about this is we know that this, under
certain conditions, will actually give
us a finite value. This will actually converge. This will actually, I guess,
give us a sensical answer. So under what conditions
does that happen? Well, this converges
if each of these terms gets smaller and
smaller and smaller. And each of these terms gets
smaller and smaller and smaller if the absolute value of our
common ratio is less than 1. So let me write that down. So this converges if
the absolute value of our common ratio
is less than 1. Or another way of
thinking about it, this is another way
of saying that x is in the interval
between-- it's less than 1 and it is greater
than negative 1. And this term right over
here, now x is a variable. x can vary between those values. We're defining a
function in terms of x. We call this the
interval of convergence. And so we know that if
x is in this interval, this is going to
give us a finite sum. And we know what
that finite sum is. It's going to be equal
to-- if it converges. So if it converges,
this is going to be equal to our first
term, which is just a-- this simplifies to a
right over here-- over 1 minus our common ratio. What's our common ratio? Our common ratio in
this example is x. Going from one term to the next,
we're just multiplying by x. We're just multiplying
by x right over there. Now, this is pretty
neat, because we're going to be able
to use this fact to put more
traditionally-defined functions into this form, and
then try to expand them out using a geometric series. And this whole idea
of using power series, or in this special
case, geometric series to represent functions,
has all sorts of applications in
engineering and finance. Using a finite number of
terms of these series, you could kind of
approximate the functions in a way that's simpler
for the human brain to understand, or
maybe a simpler way to manipulate in some way. But what's interesting
here is instead of just going from the sum
to-- instead of going from this expanded-out version
to this kind of finite value, we're now going to start
being able to take something in this form and expand it
out into a geometric series. But we have to be
careful to make sure that we're only doing it over
the interval of convergence. This is only going to be
true over the interval of convergence. Now, one other term you might
see in your mathematical career is a radius. Radius of convergence. And this is how far--
up to what value, but not including this value. So as long as our x value stays
less than a certain amount from our c value, then
this thing will converge. Now in this case,
our c value is 0. So we could ask
ourselves a question. As long as x stays
within some value of 0, this thing is going to converge. Well, you see it
right over here. As long as x stays
within one of 0. It can't go all the
way to 1, but as long as it stays less
than 1, or as long as it stays greater
than negative 1. It can stray anything
less than one away from 0, either in
the positive direction or the negative direction. Then this thing
will still converge. So we could say that our radius
of convergence is equal to 1. Another way to think about it,
our interval of convergence-- we're going from
negative 1 to 1, not including those
two boundaries, so our interval is 2. So our radius of
convergence is half of that. As long as x stays
within one of 0, and that's the same thing as
saying this right over here, this series is
going to converge.

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