Shifted Power Maps, and Integrating a Power Bound on the Derivative of a Matrix-Valued Map

In the settings of Single-Variable Calculus on an Interval and Euclidean Space and Real Matrices: Standing Notation, let $a,b\in\mathbb{R}$ with $a<b$ and let $s\in[a,b]$. By Basic Facts about Intervals of the Real Line (v), $[a,b]$ is an interval containing two distinct points, and derivatives of maps on $[a,b]$ are taken relative to $[a,b]$. For $N\in\mathbb{N}_{0}$, $N!$ is the factorial of $N$, read in $\mathbb{R}$, where $N!>0$ by Natural Numbers and Factorials Read in the Reals: the Order Carries Over, One Is Least, and Factorials Are Positive, Recursive and Nondecreasing (iii); powers in $\mathbb{R}$ are as in Powers with Exponents in the Natural Numbers with Zero; and $\psi_{N}:[a,b]\to\mathbb{R}$ is the map $r\mapsto(r-s)^{N}/N!$ of (i). Differentiability and derivatives of maps $[a,b]\to\mathbb{R}$ are as in Derivative of a Real Function on an Interval (iii) and Derivative of a Real Function on an Interval (iv), and, for $p,q\in\mathbb{N}$, of maps $[a,b]\to\mathcal{M}_{p\times q}(\mathbb{R})$ as in Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (iii) and Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (iv); continuity on $[a,b]$ is as in Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations, and $\lVert\cdot\rVert$ is the operator norm of Euclidean Space and Real Matrices: Standing Notation (vi).

(i) For every $N\in\mathbb{N}_{0}$ there is exactly one map $\psi_{N}:[a,b]\to\mathbb{R}$ with $\psi_{N}(r)=(r-s)^{N}/N!$ for every $r\in[a,b]$. $\psi_{0}(r)=1$ for every $r\in[a,b]$, and $\psi_{N}(s)=0$ for every $N\in\mathbb{N}$. For every $N\in\mathbb{N}_{0}$, $\psi_{N}$ is differentiable and continuous on $[a,b]$, and $\psi_{N+1}'(r)=\psi_{N}(r)$ for every $r\in[a,b]$.

(ii) Let $p,q\in\mathbb{N}$, let $c\in\mathbb{R}$ with $c\ge0$, let $N\in\mathbb{N}_{0}$, and let $E:[a,b]\to\mathcal{M}_{p\times q}(\mathbb{R})$ be differentiable, with derivative $E'$ continuous on $[a,b]$, such that $E(s)=0_{p\times q}$ and

$$\lVert E'(r)\rVert\le c\,\frac{|r-s|^{N}}{N!}\qquad\text{for every }r\in[a,b].$$

Then $\lVert E(t)\rVert\le c\,|t-s|^{N+1}/(N+1)!$ for every $t\in[a,b]$.