- [Instructor] So we're going to find
the radius of convergence
and the interval of convergence
for the following Taylor series.
And
this is one of our common ones,
and, actually, it's the
sum of a geometric series.
So one plus X plus X squared
plus X cubed plus dot dot dot,
and what we want is we want
to write it in summation form.
X to the N, from N
equals zero to infinity.
Now, to find...
And if you don't know this one off hand,
just do your derivatives,
add zero,
and find your terms.
Now we're going to apply the ratio test,
limit is N goes to infinity
of A sub N plus one
over A sub N.
So the limit as N goes to infinity
of X to the N plus one
over X to the N,
which actually just becomes,
this just becomes X.
So you end up with the absolute value of X
times the limit as N goes to infinity
of just one, which is one.
So we end up with the absolute value of X
and we want this to be less than one.
So right there we have
our radius of convergence.
That's a radius of convergence of one.
But let's say we wanted
to find the interval.
Then we know that X must be in
between one and negative one
and now we need to check our end points.
So first one we'll check,
let's check X equals minus one.
So we're gonna plug that in here.
If we do that,
that would give us the
series negative one to the N,
from N equals zero to infinity.
This is an alternating series
but it just alternates
between one and negative one,
which means the sum alternates
between zero and one.
We can see that the limit
as N goes to infinity
of A sub N does not equal zero,
so this is divergent,
which means we do not want
to include negative one.
Let's do the same thing for one.
If we put one in, we get one to the N,
N equals
zero to infinity,
we could use pretty much the same test.
The sum of ones is gonna go to infinity.
If we wanted to make a formal argument,
I think the easiest would be to say,
"Hey, our sequence does not go to zero.
"This diverges.
"Therefore, our interval
of convergence is,
"not including, excuse me,
"negative one to one, not
including either end point."
