I posted a question on MSE about approximating Taylor series but Despite a bounty I did not receive any answers or comments.
Maybe you guys can help.
https://math.stackexchange.com/questions/1440931/proof-that-oint-r-dx-n-n-0
Let $f(x)$ be a real-entire function such that for all $x>0$ we have $f(x) > 0$, $f'(x) > 0$ , $f '' (x) > 0$.
And also $0 < D^M f(0) < D^{M-1} f(0)$.
Let $0<T<1$ and $n$ a positive integer.
Let $g(x,n) = \frac{f(x)}{x^n}$.
Let real $x_0(n)$ satisfy $x_0 (n)> 0$
$g ' (x_0(n),n) = 0$
Let $r(n)$ be a contour that contains the real interval $[T,x_0(n) + T]$ but not $0$ , not a negative number and no poles or zeros off the real line.
Conjecture A: there is Always an integer $N$ (depending on $f$) such that for all $n > N$ $$ \frac {1} {2 \pi i} \oint_{r(n)} (x-T)^{n-1} f(x+T) - \sqrt n \ln(e+n) g(x,n) \frac{g '' (x,n) }{g ' (x,n) } < 0. $$
A similar conjecture that is probably not compatible with Conjecture A (they probably cannot both be true, but I believe at least one is true).
Conjecture B: there is Always an integer $N$ (depending on $f$) such that for all $n > N$ there exists a positive $w$ independent of $n$ such that $$ \frac {1} {2 \pi i} \oint_{r(n)} (x-T)^{n-1} f(x+T) - \frac{w g(x,n) g '' (x,n) \sqrt { f '' (x)} }{g ' (x,n) } < 0. $$
(the derivatives are with respect to $x$ , not $n$)
How to prove/disprove them ?
In case these contour integrals look random/confusing notice that
$$ \frac {1} {2 \pi i} \oint_{r(n)} (x-T)^{n-1} f(x+T) $$
Is just the sequence of Taylor coefficients for $f(x)$.
Also
$$ \frac {1} {2 \pi i} \oint_{r(n)} g(x,n) \frac{g '' (x,n) }{g ' (x,n) }$$
Is simply minimum($f(x)/x^n $) for $x > 1.$
So basically we want to understand how good the intuitive estimate min($f(x)/x^n $) is compared to $D^n f(x) / n! $.
EDIT
Maybe this helps ??