Let \(E\) be a set, and let \(f_n\colon E\to\RR\) for \(n\geq 1\) and \(f\colon E\to\RR\) be functions.
\(f_n\to f\) pointwise on \(E\) if \(\lim_{n\to\infty}f_n(x)=f(x)\) for each \(x\in E\).
\(f_n\to f\) uniformly on \(E\) if for every \(\varepsilon>0\) there exists \(N\in\NN\) such that \(\abs{f_n(x)-f(x)}<\varepsilon\) for all \(n\geq N\) and all \(x\in E\).
\(\sum_{n=1}^{\infty}f_n=f\) pointwise on \(E\) if the partial sums \(\sum_{n=1}^{N}f_n\) converge pointwise to \(f\) as \(N\to\infty\).
\(\sum_{n=1}^{\infty}f_n=f\) uniformly on \(E\) if the partial sums \(\sum_{n=1}^{N}f_n\) converge uniformly to \(f\) as \(N\to\infty\).
With the supremum norm \(\norm{g}_\infty\coloneqq\sup_{x\in E}\abs{g(x)}\) for \(g\colon E\to\RR\), condition (2) holds if and only if \(\norm{f_n-f}_\infty\to 0\).