Let the terms of a series be real functions $f_n$ defined in a certain subset $A_0$ of $\mathbb{R}$ ; we can speak of a function series. All points $x$ where the series

(1)

If the sequence $S_1,\,S_2,\,\ldots$ of the partial sums $S_n = f_1\!+\!f_2\!+\cdots+\!f_n$ of the series (1) converges uniformly in the interval $[a,\,b] \subseteq{A}$ to a function $S\!:x\mapsto S(x)$ , we say that the series converges uniformly in this interval. We may also set the direct

Definition. The function series (1), which converges in every point of the interval $[a,\,b]$ having sum function $S:x\mapsto S(x)$ , converges uniformly in the interval $[a,\,b]$ , if for every positive number $\varepsilon$ there is an integer $n_\varepsilon$ such that each value of $x$ in the interval $[a,\,b]$ satisfies the inequality $$|S_n(x)-S(x)| < \varepsilon$$ when $n \geqq n_\varepsilon$ .

The notion of uniform convergence of series can be extended to the series with complex function terms (the interval is replaced with some subset of $\mathbb{C}$ ). The significance of the uniform convergence is therein that the sum function of a series with this property and with continuous term-functions is continuous and may be integrated termwise.