- [Voiceover] Let's say that we had a function
defined by the infinite series
so we're going to from N equals one to infinity
and each term is going to be X to the N plus one
over four to the N times N to the fourth
and we could even expand this out.
When N is equal to one this is going to be
X squared over four times one to the fourth
so which is going to be four.
When X is equal to two this is going to be
X to the third power over four squared,
four squared times two to the fourth power
and we know this of course is 16 times 16
which would be 256 but I'll just leave it like that for now.
Plus when X is three, this is going to be X to the fourth
over four to the fourth power times four to the fourth
times four to the fourth which is really going to be
four to the eighth power and we'll just keep going on
and on and on just to get a sense
of what this function would look like.
This function that's defined by infinite series.
Now what I'm curious about is for what X values
will this function converge to finite values
or another way of thinking about it let's come up
with the radius of convergence
or the interval of convergence for this function.
Now since we've just been talking about the ratio test
you might say, "Well, maybe the ratio test is useful."
For what X values is this thing going to converge
so let's apply the ratio test.
Let's take the limit, the limit as N approaches infinity
of the absolute value of the N plus oneth term
over the Nth term.
For this particular case this is going to be the limit
as N approaches infinity so the N plus oneth term,
I'll do this in orange so it'll be
X to the N plus one plus one.
So it's going to be X to the N plus two,
over four to the N plus one,
times N plus one to the fourth power
and we're going to divide that by the Nth term
which is just going to be X to the N plus one
over four to the N times N to the fourth
which of course is going to be the same thing
as the limit as N approaches infinity of,
I can just copy and paste this, of this
so copy and paste that times the reciprocal of this.
We're dividing by a rational expression,
that's the same thing as multiplying by the reciprocal.
Times four to the N, N to the fourth
over X to the N plus one power.
Actually let me just clean this up a little bit.
I can re-simplify this.
Well X to the, let me get another color here
so X to the N plus two divided by X to the N plus one,
well N plus two minus N plus one is just going to be one
so this is just going to simplify to an X in the numerator.
Then let's see, four to the N divided by
four to the N plus one so this is the same thing as,
this is the same if we divide the numerator and denominator
by four to the N, this is going to be one
and this is just going to be a four.
This is going to simplify to, I don't know,
I forgot my absolute value signs, don't want to do that
so let's keep my absolute value signs here
the whole time.
This is going to be equal to the limit
as N approaches infinity of the absolute value of,
we're going to have X, N to the fourth in the numerator
so let me write that down.
We're going to have X times N to the fourth in the numerator
and in the denominator I'm going to have four
times N plus one to the fourth.
Well what's that going to be?
Well if you expand out N plus one to the fourth,
it's going to be N to the fourth
plus a bunch, a bunch of terms
and a bunch of lower degree terms,
this is going to be highest degree term
so if you distribute the four,
it's going to be four N to the fourth
plus a bunch of lower degree terms.
Let me write it that way so it's over four,
I'll use the same colors.
Four N to the fourth,
plus a bunch of lower degree terms.
Now, what's the limit as N approaches infinity here?
Well, we have the same,
we want to look at the highest degree terms.
We want to look at the N to the fourth,
the fourth degree terms
and so it's really going to be the coefficient on this
and one way to think about it if you divide the numerator
and the denominator both by N to the fourth,
these are the only terms,
you're going to have an X in the numerator
and the denominator you're going to have four
and all these others terms are going to divided
by some power of N.
As your limit approaches, as N approaches infinity
all the other terms are going to zero
and all you're going to have left
in the denominator is a four and X in the numerator.
This right over here, the limit as N approaches infinity
is going to be X over four
or I should say the absolute value of X over four.
It's going to be equal to the absolute value
of X over the absolute value of X over four.
Now we can apply the ratio test
or I guess we're always applying him
but now we can really test, do the test part of it
and the ratio test tells us
that if this limit right over here,
as long as it evaluates to something
that is less than one, then our series will converge.
We just have to say that this expression right over here
has to be less than one.
Our function is going to converge to finite values
as long as the absolute value of X over four
is less than one and this is dividing the absolute value of,
or the absolute value of X over four
is going to be the same thing as the absolute value of X
divided by four so it's four is positive
so that just has to be less than one.
Multiply both sides by four,
won't change the equality since four is positive
so you're going to have the absolute value of X
has to be less than four.
Just like that we have defined the radius of convergence.
This right over here is the radius of convergence
for our function or for our infinite series,
radius of convergence.
Or if we wanted to write that same information
as an interval of conversions, we would say
that X has to be less than four
or greater than negative four.
This is the same information right over here,
express this way, this is the interval of convergence.
These are the X values for which this original function here
is going to converge to finite values.