- [Instructor] Let's say
that we had a function
defined by the infinite series,
we're gonna go 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 it's just 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,
so it's 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 an 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.
So let's take the limit,
the limit, as n approaches infinity
of the absolute value,
the absolute value of
the n plus oneth term
over the nth term.
And so 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 gonna be x to the n plus two,
so it's gonna be x to the n plus two
over four to the n plus one,
four to the n plus one,
times n plus one to the fourth power.
And we're gonna 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.
So 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.
Now could we 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 gonna be one.
So this is just gonna simplify
to an x in the numerator.
And 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 a one.
And this is just going to be a four.
So this is going to simplify to,
and oh, I forgot my absolute value signs.
Don't wanna 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 gonna have x n to the
fourth in the numerator.
So let me write that down,
we're gonna have x times n to
the fourth in the numerator.
And in the denominator,
I'm gonna 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,
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 the
highest degree term.
So if you distribute the fourth,
it's gonna 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,
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 wanna look at the highest degree terms.
We wanna look at the n to the fourth,
the fourth degree terms.
And so it's really going to
be the coefficient on these.
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 gonna have an x in the numerator
and in the denominator,
you're gonna have a four.
and all these other terms
are gonna be divided by some power of n.
So as your limit approaches,
as n approaches infinity,
all the other terms are gonna go to zero,
and all you're gonna have left
in the denominator is a four
and x in the numerator.
So 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.
So it's going to be equal to
the absolute value of x over,
the absolute value of x over four.
And now, we can apply the ratio test,
or I guess, we were always applying here,
but now we can really
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.
And so, we just have to say
that this expression right over here
has to be less than one.
So 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.
Since four is positive,
so that just has to be less than one.
Multiply both sides by four,
won't change the inequality
since four is positive.
So you're gonna have
the absolute value of x
has to be less than four.
And so 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 convergence,
we would say that x,
x has to be less than four
or greater than negative four.
This is the same
information right over here
expressed this way,
this is the interval,
interval of convergence.
Interval of convergence.
So these are the x values
for which this original function here
is going to converge to finite values.
