Then has the characteristic property that its derivatives agree with those of the function , when both are evaluated at , up to and including the -th derivative.

Consider now the difference .

Intuitively this should be small.

There are various estimates of the size of this difference: one is the following:

So for example, for the function , we have and

When the error goes to zero as goes to infinity, we get two by-products:

- First the Taylor series converges on .
- Second the Taylor series actually represents the function on the interval .

Since this is true for any real , these Taylor series represent the functions on the entire real line.

As another example consider the function and its expansion based at 0.

We have , so, on the interval , where , we get and then we have:

Note that must be restricted to the range , since the function and its derivatives blow up as .

We conclude that the Taylor series represents the function on the interval , for any , so therefore also on the interval .

Finally, if a Taylor series converges on an open interval , then it converges absolutely on that interval.