Note that for a domain $D$, the infinite series of functions $f_n:D\to\mathbb F,$ i.e. the sum $\sum_{n=0}^\infty f_n(x)$ is nothing else as the sequence $(g_m(x))_{m\in\mathbb N}$ of its partial sums $g_m(x):=\sum_{n=0}^m f_n(x).$ We can therefore apply the notion of uniform convergence also to the members of the sequence $(g_m)_{m\in\mathbb N}.$ Clearly, by saying that $(g_m(x))_{m\in\mathbb N}$ was uniformly (or pointwise) convergent to a limit function, we mean that this limit is the function $f(x):=\sum_{n=0}^\infty f_n(x).$ Note that in this case, the terms of the infinite series $f_n$ depend on two variables - the index $n$ and the argument variable $x,$ - while the limit function $f$ only depends on the variable $x.$
The following proposition due to Karl-Theodor Weierstrass (1815 - 1897) creates a criterion for the uniform convergence in connection with an infinite series:
Let $\mathbb F$ be a either the field of real numbers or the field of complex numbers and let $D\subset \mathbb F.$ The infinite series of functions $f_n:D\to\mathbb F$ $$f(x):=\sum_{n=0}^\infty f_n(x)$$ converges absolutely and uniformly to the function $f:D\to\mathbb F,$ if the infinite series of supremum norms $$A:=\sum_{n=0}^\infty ||f_n||_\infty$$ converges to a finite limit $A < \infty.$
Proofs: 1
