|
|
|
Revision difference : Weierstrass' criterion of uniform convergence |
| Version 5 |
Version 4 |
| \begin{thmplain} |
\textbf{Theorem.} \,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 |
| $$f_1(x)+f_2(x)+...$$ |
$$f_1(x)+f_2(x)+...$$ |
| \PMlinkname{converges uniformly}{SumFunctionOfSeries} on the interval $[a, b]$. |
\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 $\mathbb{C}$. |
The theorem is valid also for the series with complex function terms, when one replaces the interval with a subset of $\mathbb{C}$. |
|
|
|
|
