- [Clark] In this video,
we're going to find
the interval of convergence
for the following power series.
And we do so, using the ratio test.
We're going to look at the
limit as k goes to infinity
of the absolute value of a sub k plus one.
So, k plus one factorial,
x to the k plus one,
all over k factorial,
x to the k.
So now, we need to rewrite this
so we can see where our cancellations are.
So, k plus one factorial
can be rewritten as
k plus one times k factorial.
And x to the k plus
one can be rewritten as
x to the k times x.
And the denominator,
not much we need to do.
We have k factorial and x to the k,
and we can see our cancels.
The k factorials will cancel
and the x to the k's will cancel.
Now, I'm going to pull the x out front,
so the absolute value of x,
and all I have left is
limit as k goes to infinity of k plus one.
Well this goes to infinity.
When this happens, our radius
of convergence is zero.
That means that it's only
convergent at a single point,
and that's what ma--
that point is whatever makes
the entire series zero.
In this case, x equals zero
is our interval of convergence.
So our interval of convergence is actually
just a single point, x equals zero.
So any time that your
limit goes to infinity,
or negative infinity, which
probably will never happen
because of the absolute value.
So any time your limit goes to infinity,
automatically, your radius
of convergence is zero
and your interval of
convergence is a single point.
And it's whatever makes
the whole thing zero
or where your power series is centered at.
