Infinite radius.
It can certainly happen that the radius of convergence is infinite.
Well, let's consider this power series.
The sum, n goes from 0 to infinity, of x to the n
divided by n factorial. Let's try the ratio test.
So, here we go.
the limit n goes to infinity of the n plus first term.
So, x to the n plus 1
over n plus 1 factorial divided by just
the nth term which is what's displayed here.
X to the n over n factorial and absolute value of
it 'because I'm checking for absolute convergence of the ratio test.
I can simplify this.
If it is a fraction with fractions in the
numerator and denominator, this is the limit n goes
to infinity of x to the n plus 1
times n factorial in the denominator of the denominator,
put that in the numerator.
Divided by x to the n times n plus 1 factorial.
So, i've just got a fraction, but I can simplify this
a bit too, this is the limit n goes to infinity.
We've got x to the n plus 1 over x to the n.
Which just leaves me with an x in the numerator, and I've got
n factor on the numerator, and an n plus 1 factorial in the denominator.
Well, n plus 1
factorial kills everything here, right?
N plus 1 factorial is 1 times 2, all the way through
n plus 1, and that contains all the terms in n factorial.
So, what I'm left with, is just an n plus 1 and the denominator.
Now, what is this limit?
X is just some fixed quantity, it doesn't depend on n.
But what's the limit then, of some number x
divided by n plus 1, n going to infinity?
Well, this limit is zero. Alright?
It doesn't matter what x is.
If you take some fixed number and divide it by a very
large quantity, you can make this as close to zero as you'd like.
So, this limit is zero. Zero is less than one.
And that means by the ratio test, this series converges regardless of what x is.
So, what's the radius of convergence? Series converges for
all values of x.
And that means the radius of convergence is infinity.
And in the not to distant future, we're going to see a very surprising result.
We're eventually going to see that this series.
Is in fact a complicated way of writing down a function we already know.
This is just a complicated way of writing down the function e to the x.
Meaning that if you plug in specific values
for x, say x equals negative 1, you'd get
that the sum n goes from 0 to infinity of minus 1 to the n over n factorial.
Is equal to e to the negative 1, it's equal to 1 over e.
And by choosing different values of x, by, by plugging in different
specific values of x, we can generate a ton of really neat series.
Like, here's another example.
I mean, just the fact that the sum, n goes from 0 to
infinity of 10 to the n over n factorial, is, well, according to this.
That's just e to the 10th power. And, that is really cool.
But, let me warn you.
And, share a bit of the philosophy of the power series with you.
Yes, by plugging in specific values of x,
you can generate a ton of interesting examples.
But, power series aren't just a way
of generating a bunch of series and isolations.
Part of the joy of power series comes by thinking of power
series, not as a, a mechanism for generating a bunch of discrete examples.
But as a way of collecting together a whole
bunch of interesting series that depend on a parameter x.
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