Bounding Taylor approximation errors for the exponential function in the presence of a power weight function

TL;DR

This paper analyzes bounds and properties of Taylor approximation errors for the exponential function with power weights, providing criteria for bounds, and establishing convexity and extremal values of related functions.

Contribution

It introduces a criterion for bounding Taylor approximation errors of the exponential function with power weights and characterizes the convexity and extremal points of associated functions.

Findings

Unique maximizers decrease from infinity to zero as delta varies.

Derived tight bounds for the maximizers s_n(delta) and u_n(delta).

Proved log-convexity of certain functions related to approximation errors.

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 $S:~\Bigl(1-e^{-s}\Bigl(1+\frac{s^1}{1!}+...+\frac{s^n}{n!}\Bigr)\Bigr)/s^{\delta} = E_{n,\delta}(s)$ 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 maximizer $s=s_n(\delta)>0$ and $u=u_n(\delta)>0$ that decrease strictly from $+\infty$ at $\delta=0$ and $\delta=n$, respectively, to 0 at $\delta=n+1$. We use Taylor's formula for truncated series with remainder in integral form to develop a criterion to decide whether a particular smooth function $S(\delta)$, $\delta\in(0,n+1)$, or $U(\delta)$, $\delta\in(n,n+1)$, respectively, is a lower/upper bound for $s_n(\delta)$ and $u_n(\delta)$, 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 $\frac{d}{d\delta}\,[{\rm ln}\,ME_{n,\delta}] ={-}{\rm ln}\,s_n(\delta)$ and $\frac{d}{d\delta}\,[{\rm ln}\,MG_{n,\delta}]={-}{\rm ln}\,u_n(\delta)$, we show that $ME_{n,\delta}$ and $MG_{n,\delta}$ are log-convex functions of $\delta\in(0,n+1)$ and $\delta\in(n+1,n)$, respectively, with limiting values 1 ($\delta\downarrow0$) and $1/(n+1)!$ ($\delta\uparrow n+1$) for $E$, and $1/n!\,(\delta\downarrow n)$ and $1/(n+1)!\,(\delta\uparrow n+1)$ for $G$. The minimal values $\hat{E}_n$ and $\hat{G}_n$ of $ME_{n,\delta}$ and $MG_{n,\delta}$, respectively, as a function of $\delta$, as well as the minimum locations $\delta_{n,E}$ and $\delta_{n,G}$ are determined in closed form.