Weierstrass' criterion of uniform convergence
WeierstrassCriterionOfUniformConvergence
2004-09-24 14:31:34
2006-09-24 02:55:26
Theorem
sum function of series
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\begin{thmplain}
\, \,Let the real functions $f_1(x)$, $f_2(x)$, ... be defined in the interval $[a, b]$. \,If they all \PMlinkescapetext{satisfy} the condition
$$|f_n(x)| \leqq M_n \quad \forall\,x\in[a, b],$$
with $\sum_{n = 1}^{\infty}M_n$ a convergent series of \PMlinkescapetext{constant terms}, then the function series
$$f_1(x)\!+\!f_2(x)\!+\!\cdots$$
\PMlinkname{converges uniformly}{SumFunctionOfSeries} on the interval $[a, b]$.
\end{thmplain}
The theorem is valid also for the series with complex function terms, when one replaces the interval with a subset of $\mathbb{C}$.