## Taylor’s Theorem

I’ve decided I really do need one convergence result for Taylor series. In the form we’ll consider today, it’s an extension of the ideas in the Fundamental Theorem of Calculus.

Recall that if the function has a continuous derivative , then the Fundamental Theorem of Calculus states that

Or, rearranging a bit

That is, we start with the value at , and we can integrate up the derivative to find how to adjust and find the value at the nearby point . Now if is *itself* continuously differentiable we can integrate by parts to find

Then we use the FToC to replace

And if is *itself* continuously differentiable we can proceed to find

Is this starting to look familiar?

At the th step we’ve got

and if is continuously differentiable we can integrate by parts and use the FToC to find

The sum is the th Taylor polynomial for — the beginning of the Taylor series of — at the point , and the integral we call the “integral remainder term” . For infinitely-differentiable functions we can define for all and get a sequence. The function is then analytic if this sequence of errors converges to in a neighborhood of .