Now if we take a look at the coefficients in front of the monomial term,

x to the n, for these three power series that we have written down.

Then we see from the recursion relation the relationship between them.

The first is the sum of the latter two.

And so if we were to subtract x times script F and

x squared times script F from script F, what would we obtain?

Most of the coefficients of the monomial terms are zero,

because of the recursion relation.

When we add up these power series, the only thing that

is non-zero is in the constant term or we get F naught and

the first order term where we get F1x- F naught x.

Now because we know the exact values of F naught and

F1, we see that, on the left, we have F- xF- x squared F.

On the right, we have simply, x.

Solving for script F gives us the power series

formula that script F, the power series,

the Fibonacci sequence is equal to x over 1- x- x squared.

Now, that's what it is, but what does it mean?

What do we do with this?

These functions that generate the power series have within them,

all sorts of interesting questions.

They tie together recursive properties, enumerative properties,

approximation properties and convergence.

We're going to focus on issues of convergence.

Consider a power series F of x, sum of a sub N, x to the n.

The following theorem holds.

For some capital R between 0 and infinity,

perhaps achieving either of those values f of x has

the following convergence behavior.

It converges absolutely, if x is less than R in absolute value,

it diverges if x is bigger than R in absolute value.

Now this R is a very special number associated to the power series.

It's called the radius of convergence.

Let's see, how would we prove this theorem?

What tool would we turn to, to determine convergence?

Well, let's start with our friend, the ratio test.

Here, what we need to do is compute rho.

The limit is n goes to infinity of the ratio,

the n plus first term to the nth term.

Now, what are the terms?

Well, here they are a sub n times x to the n.

So taking the ratio of incident terms will tell us what we need.

Now, we have to be careful to take the absolute values

since the ratio test requires a positive sequence.

Okay, so doing that what do we see?

We get some cancellation with the xs and we can rewrite this as

the limit as n goes to infinity of the absolute value of a sub n + 1 over

the absolute value of a sub n times the absolute value of x.

Now, if that is less than 1, we have absolute convergence.

If that is bigger than 1, we have divergence.

So where is this capital R coming from?

Well, it is precisely the reciprocal of this limiting coefficient.

Since if we divide by this on both sides,

we get the appropriate answer.

When rho is less than 1, that's the same thing as

saying that x is strictly less than R in absolute value.

That means we have absolute convergence.

When rho is bigger than 1 that is the same thing as saying x is

greater than R in absolute value, and that means divergence.

So what matters is this radius of convergence capital R.

The limit as n goes to infinity of absolute a sub n over a sub n+1.

Be sure to remember that, and be sure to remember that it's the nth

term over the n plus first term, not the other way.