Bounding Taylor approximation errors for the exponential function in the
presence of a power weight function
Bounding Taylor approximation errors for the exponential function in the
presence of a power weight function
Motivated by the needs in the theory of large deviations and in the theory of Lundberg's equation with heavy-tailed distribution functions, we study for $n=0,1,...$ the maximization of $S:~\Bigl(1-e^{-s}\Bigl(1+\frac{s^1}{1!}+...+\frac{s^n}{n!}\Bigr)\Bigr)/s^{\delta} = E_{n,\delta}(s)$ over $s\geq0$, with $\delta\in(0,n+1)$, $U:~({-}1)^{n+1}\Bigl(e^{-u}-\Bigl(1-\frac{u^1}{1!}+...+({-}1)^n\,\frac{u^n}{n!} \Bigr)\Bigr)/u^{\delta}=G_{n,\delta}(u)$ over $u\geq0$ with $\delta\in(n,n+1)$. We show that $E_{n,\delta}(s)$ and $G_{n,\delta}(u)$ have a unique …