- [Phillip] In this video, we're gonna
find the interval of convergence
of the following power series.
So, just like doing the
radius of convergence,
we start with our ratio test.
Limit is n goes to infinity.
Basically, ace of n plus
one, so we put in n plus one
any place we have an n, and
don't forget your parentheses.
Times the reciprocal of ace of n.
And then we're going to see
what kind of simplifying
we can do here, so we're going
to rewrite those exponents
and also rewrite the
factorials a little bit.
So first off, this is going
to be x, distributing that two
will give us to the 2n,
plus two plus another one.
So plus three, n factorial on top.
This can be rewritten as n
plus one times n factorial,
x to the 2n plus one on bottom.
So we have the limit
as n goes to infinity.
We can already see clearly
one of our cancels,
the n factorials.
Now here I have two more x's on top
than I do on bottom, so when these cancel,
I'll get left with an x squared.
Now that x squared, if we want,
we can actually pull it
out in front of the limit,
and normally, let's just
rewrite this altogether.
We can pull out the x squared,
we don't need an absolute
value because x squared
is already going to be positive.
So the limit as n goes to infinity,
and all that's left is
one over n plus one.
This is a familiar one, because we know,
as the denominator grows
and the numerator doesn't,
this is going to go to
zero, which makes this zero,
which is always going to be less than one.
Which means our radius of
convergence is infinity,
and our interval of convergence
is negative infinity
to infinity.
This is our favorite
case because that means
our power series converges
for all values of x.
