James Tanton (@jamestanton) raised the interesting question of what numbers we get if we from the infinite decimals and

Here we are placing the numbers and in the digit place in the infinite decimals.

This means that we will have to perform a lot of carries to figure out what these  numbers are.

Or does it?

What James Tanton is asking us to calculate are the numbers and

By combining two ideas of Isaac newton – calculus and power series – we can, relatively easily, calculate these numbers, and more.

They are in fact rational numbers – fractions.

Curiously, Isaac Newton thought of power series as generalizations of decimal numbers where we expand a number using the digits and powers of .

We will do just the reverse.

Let’s denote the infinite sum , where is a positive integer, by .

We will treat as a formal power series to be manipulated term-by-term, but will interpret it as a number when is chosen appropriately – mainly as .

The formal derivative, of , is the infinite sum

x

When we multiply by we get the infinite sum

That is, .

Now we can fairly easily figure out

That’s because if then

so and .

In particular,

This allows us also to calculate

In particular,

We can continue this way to calculate for higher values of .

For example so

One thing we notice is that we always get a power of 9 in the denominator.

Of course a 3 in the numerator might cancel a 3 in the denominator, so we can say that in reduced form will have a denominator that is a power of 3.