- [Instructor] In this video,
we're gonna use the ratio test
to find the interval
convergence for cosine of x.
So, the ratio test, we take
the limit as n goes to infinity
of the absolute value of a
sub n plus one over a sub n.
Which a lot of times just ends
up being a sub n plus one.
And I'll you show you something
you have to be careful about.
When you're putting in that n plus one,
it goes in for n, so you
gotta be careful about that.
A lot of people will write two n plus one
without the parentheses
and it's gonna cause an
issue with your problem.
Over two times
n plus one,
that thing factorial.
Now notice I left out the
negative one to the n.
Well, I'm taking the absolute value,
so that part's gonna disappear anyways.
And instead of dividing by a sub n,
I'm gonna multiple by one over a sub n,
because that's gonna
help with my simplifying.
So two n factorial
over x to the two n.
Right?
The rest is gonna be a
lot of simplification,
to see what we can cancel out.
So we have the limit as n goes to infinity
of the absolute value.
Now here, this is x to the two n plus two.
Down here, this is gonna be
two n plus two factorial.
This is gonna be two n factorial.
And this is gonna be x to the two n.
So let's see what all we can cancel out.
So the limit as n goes to infinity.
So I'm gonna take a step and rewrite this
before I do my canceling.
This can be rewritten as x
to the two n times x squared.
So x to the two n times x squared.
Down here, this can be
rewritten as two n plus two
times two n plus one
and then times two n factorial.
Very important piece when
doing these ratio tests
is being able to rewrite these factorials.
So hopefully you can see
where that came from.
You'd subtract one and you make a product,
and I'm not gonna go any further
because I've gone far enough
to see where my cancel is.
So the two n factorials will cancel
and the x to the two n's will cancel.
What I'm gonna do next is I'm
gonna pull the x squared out,
it does not depend on n and I
don't need the absolute value
because x squared is already positive.
So now I just need to calculate the limit
as n goes to infinity
of one over
two n plus two times two n plus one.
This limit goes to zero
so this thing is always
gonna be less than one.
Which gives me an interval of convergence
of all reals, or negative
infinity to infinity.
That's our favorite case.
If this limit goes to zero,
it tells me that this
series converges everywhere.
