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# Geometric series as a function

Video transcript

We have a function
right over here defined as an infinite series. What I want to attempt
to do in this video is to see if we can express
it in a more traditional form. And so the thing that
might jump out at you is, well, look, if this
is a geometric series, we know how to take the sum of
an infinite geometric series, at least over the x values
where the thing will actually converge. And so let's first see if
this is a geometric series. Well, the telltale pattern
of a geometric series is when you go from
one term to the next, you're multiplying
by a common ratio. So let's see. To go from 2 to
negative 8x squared, what do you have to multiply by? We have to multiply by
negative 4x squared. So we've multiplied by
negative 4x squared. Now, do we multiply
by that same amount to get to 32x to the fourth? Well sure. Negative 4x squared
times negative 8x squared is going to be positive
32x to the fourth. Multiply by negative
4x squared again, you're going to get to
negative 128x to the sixth. So the common ratio
is negative 4x squared, our first term is
2, so we can rewrite this. We can rewrite f of x as
being equal to the sum from n equals 0 to infinity
of-- let's see, our first term here is
2-- 2 times negative 4x squared to the n-th power. This is a geometric series
where our common ratio is negative 4x squared
to the n-th power. Now, when will this thing
right over here converge? Well, we know that
a geometric series will converge if the absolute
value of its common ratio is less than 1. So let me write this down. So converge if the absolute
value of the common ratio, negative 4x squared,
is less than 1. Well, this-- the way
it's written right now-- is going to be a negative value. So the absolute value of this
is just going to be 4x squared. Right? x squared is going
to be non-negative, so 4x squared is going
to be non-negative. Negative 4x squared is
going to be non-positive. So if you're taking the absolute
value of a non-positive thing, that's going to be the same
thing as the absolute value of the negative of it. So this just has
to be less than 1. And the absolute
value of something that is strictly
non-negative like this, well, that's just going
to be 4x squared-- these two statements
are equivalent-- and that has to be less than 1. Can divide both sides by 4, you
get x squared is less than 1/4. And so we can say that
the absolute value of x has to be less than 1/4, or
we could say that negative 1/4 has to be less than x, which has
to be less than positive 1/4. So expressed this
way, we're giving the interval of convergence. This thing will converge as
long as x is in this interval. Expressed this way,
we're really saying the radius of convergence. This will converge
as long as x is less than our radius
of convergence, as long as the
absolute value of x is less than our
radius of convergence, as long as x stays less
than 1/4 away from 0. To make it a little
bit clearer, you could rewrite this as the
distance between x and 0, as long as this- this,
you could view this as the distance
between x and 0-- as long as this
stays less than 1/4, this thing is going to converge. So this is the interval
of convergence, this, you could view,
1/4, you could view as the radius of convergence. Now with that out
of the way, we've thought about where
this thing converges, let's think about
what it converges to. Well, we've done
this multiple times. This is going to be equal
to our first term, 2 over 1 minus our common ratio. Our common ratio is
negative 4x squared. So this is going
to give us-- and we deserve a drum roll-- 2
over 1 plus 4x squared. So this expression is
going to be equal to this, as long as x is within our
interval of convergence.