A series of real-valued functions and its pointwise and uniform convergence through its partial sums

Let $X$ be a set and let $f_k:X\to\mathbb{R}$ for $k\in\mathbb{N}$. The series of real-valued functions $\sum f_k$ is studied through its partial-sum functions

$S_n(x):=\sum_{k<n}f_k(x)\qquad(n\in\mathbb{N},\ x\in X),$

where the sum on the right is the finite sum of Series, partial sums, convergence and the sum, divergence, and the tail series. Thus $S_0$ is the zero function and $S_{n+1}=S_n+f_n$ under the pointwise operations of The vector space $F^{X}$ of all functions $X \to F$ with pointwise operations, and $F^{n}$ as the case $X = n = \{0, 1, \dots, n-1\}$.

The series $\sum f_k$ converges pointwise to $S:X\to\mathbb{R}$ when $S_n\to S$ pointwise, and it converges uniformly to $S$ when $S_n\to S$ uniformly (Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions).

The series is absolutely convergent at $x\in X$ when the scalar series $\sum |f_k(x)|$ converges. It is absolutely pointwise convergent when this holds for every $x\in X$.