- [Narrator] In this
video we're gonna find the
interval of convergence
for the following power
series, and we do so by
using the ratio test.
So the limit as k goes to infinity
of a sub k plus one,
so x to the k plus one,
over, now you need to be
a little careful here.
I'm gonna do it in two steps.
K plus one in parentheses,
the factorial on the outside.
Now instead of dividing by a sub k,
we'll just multiply by the reciprocal.
So 2k factorial over x to the k.
And let's see how this
one is going to simplify.
So we have the limit
as k goes to infinity.
This is gonna be x to the k times x,
and this is gonna stay two k
factorial on top.
On bottom,
notice we have two k plus two factorial.
Now what would happen if
I didn't put the k plus
one in parentheses, or
I see a lot, people will
just write two k plus one
factorial, and it's not
quite correct. Although
in this particular case
it's not really gonna affect our answer.
X to the k.
Now we need another
step of simplification.
So we have the limit
as k goes to infinity.
X to the k. X.
Two k factorial.
What we need to work on is
rewriting this factorial.
Because we want to get
that two k factorial
canceled out.
Well two k factorial,
the first term is two k
plus two. The next term is two k plus one.
The next term would be two
k, and then so on down.
So that's where it becomes
the two k factorial.
And then x to the k.
This allows us to very
clearly see our
cancellations. The x to the ks
will cancel, and now the two k
factorials will cancel.
X is all that's left that
does not depend on k,
and I'll pull that out
front along with the
absolute value.
And I've got the limit
as k goes to infinity
of just one left on
top over two k plus two
times two k plus one,
and I don't think there's
any reason to FOIL out that denominator
because we can see
what's gonna happen, this
denominator's gonna grow,
the numerator's not.
This limit is gonna go to zero
which automatically, because x is
some number, is gonna make this zero,
which is always less than one.
This gives us an infinite
radius of convergence,
and interval of convergence
of negative infinity
to infinity. This truly
is our favorite case for
power series, because it
means that our power series
converges for all values
of x, and we don't have
to test the end points. We're done very
quickly, so if that limit goes to zero,
automatically your
radius of convergence is
infinity, and your interval
of convergence is all
real numbers.
