Weierstrass’ criterion of uniform convergence

Theorem.

Let the real functions $f_1(x)$, $f_2(x)$, … be defined in the interval $[a,b]$.  If they all the condition

$$|f_n(x)|\leqq M_n\mathit{\hspace{1em}}\forall x\in [a,b],$$

with ${\sum }_{n=1}^{\mathrm{\infty }}{M}_n$ a convergent series of , then the function series

$$f_1(x)+f_2(x)+\mathrm{\cdots }$$

converges uniformly (http://planetmath.org/SumFunctionOfSeries) on the interval $[a,b]$.

The theorem is valid also for the series with complex function terms, when one replaces the interval with a subset of $ℂ$.