5.9.E: Problemas de Convergencia en Diferenciación e Integración
- Page ID
- 114008
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Completar todos los detalles de prueba en Teoremas 1 y\(3,\) Corolarios 1 y\(2,\) y Nota\(3 .\)
Demostrar que las suposiciones (a) y (c) en el Teorema 1 pueden ser reemplazadas por\(F_{n} \rightarrow F\) (puntualmente) on\(I\). (De esta forma, el teorema se aplica\(E\) también a espacios incompletos.)
\(\left.\text { [Hint: } F_{n} \rightarrow F(\text { pointwise } e), \text { together with formula ( } 3\right),\)implica\(F_{n} \rightarrow F\) (uniformemente)\(\text { on } I .]\)
Mostrar que el Teorema 1 falla sin suposición\((\mathrm{b}),\) incluso si\(F_{n} \rightarrow F\) (uniformemente) y si\(F\) es diferenciable en\(I .\)
[Pista: Para un contraejemplo, pruebe\(F_{n}(x)=\frac{1}{n} \sin n x,\) cualquier no degenerado\(I .\) Verifique que\(F_{n} \rightarrow 0\) (uniformemente), sin embargo (b) y aserción (iii) fallan.
Demostrar el teorema de Abel (\(4, §13,\)Problema del Capítulo 15) para la serie
\ [\ suma a_ {n} (x-p) ^ {n},
\]
con todo\(a_{n}\) en\(E^{m}\left(^{*} \text { or in } C^{m}\right)\) pero con\(x, p \in E^{1}\).
[Pista: Dividir\(a_{n}(x-p)^{n}\) en componentes.]
Demostrar Corolario 3.
\(\text { [Hint: By Abel's theorem (see Problem } 4),\)podemos poner
\ [
\ sum_ {n=0} ^ {\ infty} a_ {n} (x-p) ^ {n} =F (x)
\]
\(\left.\text { uniformly on }\left[p, x_{0}\right] \text { (respectively, }\left[x_{0}, p\right]\right) .\) Esto implica que\(F\) es relativamente continuo en\(x_{0} .\) (¿Por qué?) Así es\(f,\) por suposición. También\(f=F\) en\(\left[p, x_{0}\right)\left(\left(x_{0}, p\right]\right) .\) Mostrar que
\ [
f\ left (x_ {0}\ right) =\ lim f (x) =\ lim F (x) =F\ left (x_ {0}\ right)
\]
como\(x \rightarrow x_{0}\) desde la izquierda (derecha).]
En los siguientes casos, encuentre la serie Taylor de\(F\) aproximadamente 0 integrando la serie de Teorema de\(F^{\prime} .\) Uso 3 y Corolario 3 para encontrar el radio de convergencia\(r\) e investigar la convergencia en\(-r\) y\(r .\) Usar\((\mathrm{b})\) para encontrar una fórmula para\(\pi .\)
(a) \(F(x)=\ln (1+x)\);
b)\(F(x)=\arctan x\);
c)\(F(x)=\arcsin x\).
Demostrar que
\ [\ int_ {0} ^ {x}\ frac {\ ln (1-t)} {t} d t=\ sum_ {n=1} ^ {\ infty}\ frac {x^ {n}} {n^ {2}}\ quad\ texto {para} x\ in [-1,1].
\]
[Pista: Usar Teorema 3 y Corolario\(3 .\) Tomar derivadas de ambos lados.]