Taylor’s Theorem again
What you say? Taylor’s Theorem again? Well, yes. But not quite. Now instead of showing it as an extension of the Fundamental Theorem of Calculus we’ll see it’s an extension of the Differential Mean Value Theorem. And this despite the most strenuous objections of my resident gadfly that the DMVT and FToC have absolutely nothing whatsoever to do with each other, never have, and never will.
Remember that Taylor’s theorem tells us that if is times continuously differentiable we can write
where the remainder term is given in the integral form
What I want to say today is that it can also be given in a different form. Specifically, there is some between and so that
Now, we could actually weaken the assumptions on slightly and work up a whole new induction from the ground up using a souped-up version of the DMVT. In particular, notice that the case says there is a between and so that
which is just a different way of writing the DMVT itself.
Instead of doing all that hard work, I’ll prove that this form of the remainder term holds under the same conditions as the integral version. And I’ll do this by using a form of the Integral Mean Value Theorem for Riemann-Stieltjes integration.
Specifically, we’ve got an integral to evaluate
and antidifferentiate to find
Now the IMVT tells us that there is a between and so that
Again, if we can make converge to in a neighborhood of , then the infinitely-differentiable function is actually analytic. Here the limit gets fidgety because we don’t know where between and we might pick , and the jumps around as we use larger and larger . But if we can keep good control on the size of all of the derivatives of near , this can be a useful strategy.