Bounding Taylor approximation errors for the exponential function in the presence of a power weight function
A. J. E. M. Janssen
Abstract
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 $S:~\Bigl(1-e^{-s}\Bigl(1+\frac{s^1}{1!}+...+\frac{s^n}{n!}\Bigr)\Bigr)/s^δ = E_{n,δ}(s)$ over $s\geq0$, with $δ\in(0,n+1)$, $U:~({-}1)^{n+1}\Bigl(e^{-u}-\Bigl(1-\frac{u^1}{1!}+...+({-}1)^n\,\frac{u^n}{n!} \Bigr)\Bigr)/u^δ=G_{n,δ}(u)$ over $u\geq0$ with $δ\in(n,n+1)$. We show that $E_{n,δ}(s)$ and $G_{n,δ}(u)$ have a unique maximizer $s=s_n(δ)>0$ and $u=u_n(δ)>0$ that decrease strictly from $+\infty$ at $δ=0$ and $δ=n$, respectively, to 0 at $δ=n+1$. We use Taylor's formula for truncated series with remainder in integral form to develop a criterion to decide whether a particular smooth function $S(δ)$, $δ\in(0,n+1)$, or $U(δ)$, $δ\in(n,n+1)$, respectively, is a lower/upper bound for $s_n(δ)$ and $u_n(δ)$, respectively. This criterion allows us to find lower and upper bounds for $s_n$ and $u_n$ that are reasonably tight and simple at the same time. Furthermore, as a consequence of the identities $\frac{d}{dδ}\,[{\rm ln}\,ME_{n,δ}] ={-}{\rm ln}\,s_n(δ)$ and $\frac{d}{dδ}\,[{\rm ln}\,MG_{n,δ}]={-}{\rm ln}\,u_n(δ)$, we show that $ME_{n,δ}$ and $MG_{n,δ}$ are log-convex functions of $δ\in(0,n+1)$ and $δ\in(n+1,n)$, respectively, with limiting values 1 ($δ\downarrow0$) and $1/(n+1)!$ ($δ\uparrow n+1$) for $E$, and $1/n!\,(δ\downarrow n)$ and $1/(n+1)!\,(δ\uparrow n+1)$ for $G$. The minimal values $\hat{E}_n$ and $\hat{G}_n$ of $ME_{n,δ}$ and $MG_{n,δ}$, respectively, as a function of $δ$, as well as the minimum locations $δ_{n,E}$ and $δ_{n,G}$ are determined in closed form.