- [Phillip] So we're trying to find
the radius of convergence of
the following power series,
and what we do is we first
apply the ratio test.
The limit as n goes to infinity of,
just as a reminder, a-sub-n plus one
over a-sub-n, the absolute value of that,
and so setting this up,
basically what we end up doing is
a-sub-n plus one times
the reciprocal of a-sub-n
as a bit of a shortcut.
So limit as n goes to infinity.
Let's start with a-sub-n plus one.
So that'll be two
times n plus one
factorial
x to the n plus one
over
n plus one
factorial squared,
and we really need to pay
attention to our parentheses and
where the plus one goes.
Now times the reciprocal of our original.
N factorial
squared
over
two n
factorial
x to the n.
Okay.
Now, we'll do a little bit of rearranging,
and then we're going to do all,
the rest will be algebra, so.
Here, this'll be,
oops, let's,
let's put in our limit
as n goes to infinity.
Two n plus two
factorial.
This one, I'm just going to write as,
n plus one factorial, we can rewrite as
n plus one
times n
factorial
squared.
So I'm rewriting
the n plus one factorial there.
May be hard to read,
that's a
factorial symbol.
And here we've got
n factorial
squared,
two n
factorial, x to the n.
Now we'll start to see what kind of
expansions and canceling we can do.
(paper rustling)
So with factorials, what
we're going to want to
is write those out as much as possible.
So two n,
or at least until we see it cancels.
Two n plus two factorial
is
two n plus two,
times two n plus one,
times
two n
factorial.
And now we see that
our cancel there.
Here, I'm going to distribute to square.
So I have an n plus one squared,
and
and n factorial squared.
And we see another cancel,
and somewhere along the lines
I lost my x to the n plus one here.
And I'm about to lose it again here.
X to the n
times x.
So I'm going to rewrite
that x to the n plus one
as x to the n times x.
N factorial squared,
over
two n factorial x to the n.
So we can start doing some canceling.
The x to the n's are going to cancel.
The n factorial squareds
are going to cancel,
and the two n factorials
are going to cancel.
The next thing we'll do,
is we'll pull out anything
that doesn't depend on n,
which in this case is just this x.
And I'm going to write it
out in front of the limit.
So the absolute value of x,
limit as
n goes to infinity.
I no longer need the absolute value here
because this is all going to be positive.
On top I've got two n plus two,
times two n plus one,
over
n plus one the quantity squared.
So I've got the absolute value of x.
Now, this limit,
we're going to have a
quadratic over a quadratic,
and we'll be able to show really quickly
with L'Hôpital's Rule, that this limit
is just going to be four.
Now, what we want is for
this to be less than one,
in order for this to converge.
So,
in solving this, what we get
is that the absolute value of x
has to be less than 1/4.
So what we've got in this 1/4
is our radius of convergence.
That completes this problem.
(heavy breathing)
