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Revision difference : Weierstrass' criterion of uniform convergence |
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Version 5 |
| \begin{thmplain} |
\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 |
\, \,Let the real functions $f_1(x)$, $f_2(x)$, ... be defined in the interval $[a, b]$. \,If they all \PMlinkescapetext{satisfy} the condition |
| $$|f_n(x)| \leqq M_n \quad \forall\,x\in[a, b],$$ |
$$|f_n(x)| \leqq M_n \quad \forall\,x\in[a, b],$$ |
| with $\sum_{n = 1}^{\infty}M_n$ a convergent series of \PMlinkescapetext{constant terms}, then the function series |
with $\sum_{n = 1}^{\infty}M_n$ a convergent series of \PMlinkescapetext{constant terms}, then the function series |
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$$f_1(x)\!+\!f_2(x)\!+\!\cdots$$
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$$f_1(x)+f_2(x)+...$$
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| \PMlinkname{converges uniformly}{SumFunctionOfSeries} on the interval $[a, b]$. |
\PMlinkname{converges uniformly}{SumFunctionOfSeries} on the interval $[a, b]$. |
| \end{thmplain} |
\end{thmplain} |
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| The theorem is valid also for the series with complex function terms, when one replaces the interval with a subset of $\mathbb{C}$. |
The theorem is valid also for the series with complex function terms, when one replaces the interval with a subset of $\mathbb{C}$. |
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