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 .