- In this video, we're
gonna use the Ratio Test
to determine the interval
of convergence for
the Taylor series one over one minus x
and the nth degree Taylor
polynomial for this function is
k equals zero to n of x to the k.
So it's Taylor Series,
we can use n, or which
ever letter we want,
n to infinity of x to the n.
So here's what we're gonna
apply the Ratio Test to.
For the Ratio Test, we get the
limit, as n goes to infinity,
of the absolute value of a sub n plus one
divided by a sub n, so x to the n plus one
over x to the n.
This one actually simplifies quite easily
because what we end up with
is the limit, as n goes to infinity,
of just x, the absolute value of x.
which is just the absolute value of x.
So we get that the absolute value of x
must be less than one.
Now, what we'll do in this case
is now test the end points to see
what our interval of convergence is,
because right now, what we
know is that x has to be
in between negative one and one,
but because the Ratio Test is inconclusive
when x equals one, those
must be tested separately.
So, if
x equals negative one,
then our series becomes
n equals zero to infinity
of negative one to the n,
which is a series that
just bounces back and forth
between one and negative one.
And so,
the limit, as n goes to infinity,
of negative one to the
n does not equal zero
therefore it's divergent
by the Divergence Tests.
So we do not want to include negative one.
If x equals positive one,
we get the summation,
from n equals zero to
infinity, of one to the n,
which is summation that n equals
zero to infinity of just one.
We can actually use the
same exact argument.
The limit, as n goes to infinity,
of one, does not equal zero.
So it's again, divergent
by the Divergence Test.
So our interval of convergence
is negative one to one
not including either end point.
So our interval of convergence,
if we want to write it in
interval notation: negative one to one.
