Iterated Powers, Factorials, and Convergence of the Series of Powers over Factorials

In the setting of The Real Numbers: Standing Notation and Background, let $A$ be a real number with $0\le A$. For $k\in\mathbb{N}$ let $k!$ denote the factorial of $k$, which by the finite-product definition used there is a real number, and let $c^{n}$ denote the $n$th power of a real number $c$. Convergence of a series of real numbers and its sum are as defined there. Then the following hold.

1. (Doubling, iterated powers, and absolute values)(i) Let the multiplication of the natural numbers be that of clauses 3 and 4 of that definition, let $S$ be its successor map, and for $k\in\mathbb{N}$ abbreviate

$$2k=k\cdot 2,\qquad 2k+1=S(k\cdot 2).$$

Then $2k=k+k$ and $k<2k$. Moreover, for every real number $c$ and all $m,n\in\mathbb{N}$,

$$c^{\,m\cdot n}=\bigl(c^{m}\bigr)^{n}, \qquad \bigl|c^{n}\bigr|=|c|^{n},$$

where $|t|$ denotes the absolute value of $t\in\mathbb{R}$. In particular $(-c)^{2n}=c^{2n}$ and $(-c)^{2n+1}=-c^{2n+1}$.

2. (Factorials are positive and nondecreasing)(ii) For every $k\in\mathbb{N}$ one has $1\le k!$; in particular $k!$ is positive and has a multiplicative inverse in $\mathbb{R}$. Moreover $m!\le n!$ for all $m,n\in\mathbb{N}$ with $m\le n$.

3. (The exponential majorant)(iii) The terms of the series $\sum_{k=1}^{\infty}\dfrac{A^{k}}{k!}$ are nonnegative, and that series converges.

4. (The trigonometric majorants)(iv) For every $k\in\mathbb{N}$,

$$0\le\frac{A^{2k}}{(2k)!}\le\frac{(A^{2})^{k}}{k!}, \qquad 0\le\frac{A^{2k+1}}{(2k+1)!}\le A\cdot\frac{(A^{2})^{k}}{k!} .$$

Consequently the series

$$\sum_{k=1}^{\infty}\frac{A^{2k}}{(2k)!} \qquad\text{and}\qquad \sum_{k=1}^{\infty}\frac{A^{2k+1}}{(2k+1)!}$$

converge.