Series of Real-Valued Functions and Their Partial Sums

In the setting of The Real Numbers: Standing Notation and Background, let $D$ be a set, let $S\subseteq D$, let $(g_{k})_{k\in\mathbb{N}}$ be a sequence in the set of functions from $D$ to $\mathbb{R}$, and let $f:D\to\mathbb{R}$.

1. (Partial sums)(i) The partial sums of $(g_{k})_{k\in\mathbb{N}}$ are the functions $s_{m}:D\to\mathbb{R}$, one for each $m\in\mathbb{N}$, given by the finite sum

$$s_{m}(x)=\sum_{k=1}^{m}g_{k}(x)\qquad(x\in D),$$

and $(s_{m})_{m\in\mathbb{N}}$ is a sequence in the set of functions from $D$ to $\mathbb{R}$.

2. (Pointwise convergence of the series)(ii) The series $\sum_{k=1}^{\infty}g_{k}$ converges pointwise to $f$ on $S$ if the sequence of partial sums $(s_{m})_{m\in\mathbb{N}}$ converges pointwise to $f$ on $S$.

3. (Uniform convergence of the series)(iii) The series $\sum_{k=1}^{\infty}g_{k}$ converges uniformly to $f$ on $S$ if the sequence of partial sums $(s_{m})_{m\in\mathbb{N}}$ converges uniformly to $f$ on $S$.