- [Instructor] In this video,
we're gonna apply the ratio test
to find out the interval of
convergence for sine of X,
whose Taylor series is given
by an equals zero to infinity
of negative one to the
N, X to the 2N plus one,
or 2N plus one factorial.
I'm gonna go right into
applying the ratio test,
so limit is N goes to infinity
of X to the N plus one,
and it's an absolute value,
so you don't need to worry
about the alternating piece.
And so X to the two, now remember,
you're putting in N plus one for N,
so it's a good habit to put in parentheses
because I see people just
add one onto the end,
and it will mess up your canceling.
So two times N plus
one plus one factorial.
And instead of dividing by A sub N,
I like to just think of it as
multiplying by the reciprocal,
because I'm gonna be looking
for canceling anyways.
So that just kinda cuts out a step for me.
All right, let's rewrite these,
simplify all the parentheses,
and then we'll start to look
at what we can factor and cancel.
So this is gonna be 2N
plus two, plus another one,
so that's gonna be a 2N plus three.
So X to the 2N plus three here.
Over 2N plus three factorial
times 2N plus one factorial
over X to the 2N plus one.
All right, so we got the
limit is N goes to infinity.
Now we're gonna look
to see what'll factor.
Now for instance, this,
this is basically 2N plus
three Xs multiplied together,
so I can rewrite that
as X to the 2N plus one,
and times X squared, so I'm
splitting off two of 'em
to make it match up with this term here.
I'm gonna do something similar here.
I'm gonna start to write out
the product of this factorial,
so 2N plus three times 2N plus two,
and I'm gonna stop
there and write the rest
as 2N plus one factorial.
Once again, to match up
with this 2N plus one factorial here,
and X to the 2N plus one.
Now I can clearly see my canceling.
Now I'm gonna pull out the X squared,
and I no longer need the absolute value
'cause X squared's
always gonna be positive,
so I'm not worried about that.
And the only part that depends on N,
limit is N goes to infinity
of one over 2N plus three
times 2N plus two, and I can see without,
I've got one over, basically
something is going to infinity,
which means this equals zero
which means it's always
gonna be less than one.
This is our favorite case,
which means we have a radius
of convergence of infinity,
and our interval of convergence
is negative infinity to infinity,
which means this Taylor series converges
for all values of X.
