Taylor series: criteria for maximum error attained at boundary? In Calculus BC there's a frequent assumption that the error between an $n$-th degree Taylor series and its corresponding function is larger the further the functions are evaluated from the Taylor series' center.
It's not quite true. You can provide counterexamples using polynomials. However, I was wondering if there's a subtler set of criteria where it works. It seems to hold for $\sin x,\ \cos x,\ \text{and}\ e^x$. 
On the complex plane, if our function is analytic, then the difference of the function and its Taylor series is also analytic, so the maximum modulus of the difference occurs on the boundary (of a reasonably defined domain). So I'm wondering if there are any similar criteria where we could say, over the domain of real numbers, for any ball whose center is the center of the Taylor series, the extrema of the difference between the Taylor series and its original function occurs at the boundary.
 A: If the "boundary" is defined arbitrarily, then of course we can expect examples of both kinds: (a) the Taylor series has the largest error at a point of the "boundary"; (b) the Taylor series has a larger error in a certain point in the "interior" than at any point of the "boundary". Both examples might be the case if the domain of our function is $[a,b] \subset {\mathbb R}$ and the "boundary" consists of the two points $a$ and $b$.
But it is more natural to study the convergence of power series on the complex plane (i.e. consider our function to be an analytic function of a complex variable). 
So, if our function is analytic and if we define the "boundary" as the boundary of the disk of convergence on the complex plane (i.e. a disk whose radius is the radius of convergence), then we know that there exists a singularity at the boundary (assuming the radius of convergence is finite). It is intuitively clear that the error grows much faster in the neighborhood of the singularity than at most other points (far from the singularity) within the disk of convergence.
