Term-by-Term Differentiation of a Series of Continuously Differentiable Functions with Summable Uniform Bounds

Adopt Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation and fix a dimension $q$; the set $\mathbb{R}^{q}$ is open by claim 1 of Euclidean Space Is Open in Itself. Convergence and absolute convergence of a series of real numbers, and the sum $\sum_{k=1}^{\infty}$ of a convergent series, are those of that definition. Let $(f_{k})_{k\in\mathbb{N}}$ be a sequence of functions $\mathbb{R}^{q}\to\mathbb{R}$, each of class $C^{1}$ on $\mathbb{R}^{q}$, and let $(a_{k})_{k\in\mathbb{N}}$ and $(b_{k})_{k\in\mathbb{N}}$ be sequences of nonnegative real numbers whose series $\sum_{k=1}^{\infty}a_{k}$ and $\sum_{k=1}^{\infty}b_{k}$ converge, such that

$$|f_{k}(x)|\le a_{k}\qquad\text{and}\qquad|\partial_{i}f_{k}(x)|\le b_{k}\qquad\text{for all }k\in\mathbb{N},\ x\in\mathbb{R}^{q}\text{ and }i\in[q].$$

Then the following hold.

1. (Convergence)(i) For every $x\in\mathbb{R}^{q}$ and every $i\in[q]$ the series $\sum_{k=1}^{\infty}f_{k}(x)$ and $\sum_{k=1}^{\infty}\partial_{i}f_{k}(x)$ converge absolutely. Let $F:\mathbb{R}^{q}\to\mathbb{R}$ and $G_{i}:\mathbb{R}^{q}\to\mathbb{R}$ be their sums, $F(x)=\sum_{k=1}^{\infty}f_{k}(x)$ and $G_{i}(x)=\sum_{k=1}^{\infty}\partial_{i}f_{k}(x)$.

2. (Term-by-term differentiation)(ii) $F$ is of class $C^{1}$ on $\mathbb{R}^{q}$ with $\partial_{i}F=G_{i}$ for every $i\in[q]$, and

$$|F(x)|\le\sum_{k=1}^{\infty}a_{k}\qquad\text{and}\qquad|\partial_{i}F(x)|\le\sum_{k=1}^{\infty}b_{k}\qquad\text{for all }x\in\mathbb{R}^{q}\text{ and }i\in[q].$$