1.2: Preparación
- Page ID
- 109234
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Comenzamos con dos criterios de convergencia útiles para integrales inadecuados que no involucren un parámetro. Consistente con la definición de la p. 152, decimos que\(f\) es integrable localmente en un intervalo\(I\) si es integrable en cada subintervalo cerrado finito de\(I\).
[teorem:2] Supongamos que\(g\) es integrable localmente\([a,b)\) y denota
\[G(r)=\int_{a}^{r}g(x)\,dx,\quad a\le r<b.\]
Entonces la integral impropia\(\int_{a}^{b}g(x)\,dx\) converge si y solo si\(,\) para cada uno\(\epsilon >0,\) hay\(r_{0}\in[a,b)\) tal que
\[\label{eq:9} |G(r)-G(r_{1})|<\epsilon,\quad r_{0}\le r,r_{1}<b.\]
Por necesidad, supongamos\(\int_{a}^{b}g(x)\,dx=L\). Por definición, esto quiere decir que para cada uno\(\epsilon>0\) hay\(r_{0}\in [a,b)\) tal que
\[|G(r)-L|<\frac{\epsilon}{2} \text{\quad and\quad} |G(r_{1})-L|<\frac{\epsilon}{2},\quad r_{0}\le r,r_{1}<b.\]
Por lo tanto
\[\begin{aligned} |G(r)-G(r_{1})|&=&|(G(r)-L)-(G(r_{1})-L)|\\ &\le& |G(r)-L|+|G(r_{1})-L|< \epsilon,\quad r_{0}\le r,r_{1}<b.\end{aligned}\]
Para la suficiencia, [eq:9] implica que
\[|G(r)|= |G(r_{1})+(G(r)-G(r_{1}))|< |G(r_{1})|+|G(r)-G(r_{1})|\le |G(r_{1})|+\epsilon,\]
\(r_{0}\le r\le r_{1}<b\). Dado que también\(G\) está acotado en el conjunto compacto\([a,r_{0}]\) (Teorema 5.2.11, p. 313),\(G\) está acotado en\([a,b)\). Por lo tanto, las funciones monótona
\[\overline{G}(r)=\sup\left\{G(r_{1})\, \big|\, r\le r_{1}<b\right\} \text{\quad and\quad} \underline{G}(r)=\inf\left\{G(r_{1})\, \big|\, r\le r_{1}<b\right\}\]
están bien definidos en\([a,b)\), y
\[\lim_{r\to b-}\overline{G}(r)=\overline{L} \text{\quad and\quad} \lim_{r\to b-}\underline{G}(r)=\underline{L}\]
ambos existen y son finitos (Teorema 2.1.11, p. 47). De [eq:9],
\[\begin{aligned} |G(r)-G(r_{1})|&=&|(G(r)-G(r_{0}))-(G(r_{1})-G(r_{0}))|\\ &\le &|G(r)-G(r_{0})|+|G(r_{1})-G(r_{0})|< 2\epsilon,\end{aligned}\]
por lo
\[\overline{G}(r)-\underline{G}(r)\le 2\epsilon, \quad r_{0}\le r, r_{1}<b.\]
Dado que\(\epsilon\) es un número positivo arbitrario, esto implica que
\[\lim_{r\to b-}(\overline{G}(r)-\underline{G}(r))=0,\]
así\(\overline{L}=\underline{L}\). Vamos\(L=\overline{L}=\underline{L}\). Desde
\[\underline{G}(r)\le G(r)\le \overline{G}(r),\]
de ello se deduce\(\lim_{r\to b-} G(r)=L\).
Te dejamos la prueba del siguiente teorema (Ejercicio [exer:2]).
[teorem:3] Supongamos que\(g\) es integrable localmente\((a,b]\) y denota
\[G(r)=\int_{r}^{b}g(x)\,dx,\quad a\le r<b.\]
Entonces la integral impropia\(\int_{a}^{b}g(x)\,dx\) converge si y solo si\(,\) para cada uno\(\epsilon >0,\) hay\(r_{0}\in(a,b]\) tal que
\[|G(r)-G(r_{1})|<\epsilon,\quad a<r,r_{1}\le r_{0}.\]
Para ver por qué asociamos Teoremas [teorem:2] y [teorem:3] con Cauchy, compararlos con Teorema 4.3.5 (p. 204)