1.2: Preparation
- Page ID
- 17325
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We begin with two useful convergence criteria for improper integrals that do not involve a parameter. Consistent with the definition on p. 152, we say that \(f\) is locally integrable on an interval \(I\) if it is integrable on every finite closed subinterval of \(I\).
[theorem:2] Suppose \(g\) is locally integrable on \([a,b)\) and denote
\[G(r)=\int_{a}^{r}g(x)\,dx,\quad a\le r<b.\]
Then the improper integral \(\int_{a}^{b}g(x)\,dx\) converges if and only if\(,\) for each \(\epsilon >0,\) there is an \(r_{0}\in[a,b)\) such that
\[\label{eq:9} |G(r)-G(r_{1})|<\epsilon,\quad r_{0}\le r,r_{1}<b.\]
For necessity, suppose \(\int_{a}^{b}g(x)\,dx=L\). By definition, this means that for each \(\epsilon>0\) there is an \(r_{0}\in [a,b)\) such that
\[|G(r)-L|<\frac{\epsilon}{2} \text{\quad and\quad} |G(r_{1})-L|<\frac{\epsilon}{2},\quad r_{0}\le r,r_{1}<b.\]
Therefore
\[\begin{aligned} |G(r)-G(r_{1})|&=&|(G(r)-L)-(G(r_{1})-L)|\\ &\le& |G(r)-L|+|G(r_{1})-L|< \epsilon,\quad r_{0}\le r,r_{1}<b.\end{aligned}\]
For sufficiency, [eq:9] implies that
\[|G(r)|= |G(r_{1})+(G(r)-G(r_{1}))|< |G(r_{1})|+|G(r)-G(r_{1})|\le |G(r_{1})|+\epsilon,\]
\(r_{0}\le r\le r_{1}<b\). Since \(G\) is also bounded on the compact set \([a,r_{0}]\) (Theorem 5.2.11, p. 313), \(G\) is bounded on \([a,b)\). Therefore the monotonic functions
\[\overline{G}(r)=\sup\left\{G(r_{1})\, \big|\, r\le r_{1}<b\right\} \text{\quad and\quad} \underline{G}(r)=\inf\left\{G(r_{1})\, \big|\, r\le r_{1}<b\right\}\]
are well defined on \([a,b)\), and
\[\lim_{r\to b-}\overline{G}(r)=\overline{L} \text{\quad and\quad} \lim_{r\to b-}\underline{G}(r)=\underline{L}\]
both exist and are finite (Theorem 2.1.11, p. 47). From [eq:9],
\[\begin{aligned} |G(r)-G(r_{1})|&=&|(G(r)-G(r_{0}))-(G(r_{1})-G(r_{0}))|\\ &\le &|G(r)-G(r_{0})|+|G(r_{1})-G(r_{0})|< 2\epsilon,\end{aligned}\]
so
\[\overline{G}(r)-\underline{G}(r)\le 2\epsilon, \quad r_{0}\le r, r_{1}<b.\]
Since \(\epsilon\) is an arbitrary positive number, this implies that
\[\lim_{r\to b-}(\overline{G}(r)-\underline{G}(r))=0,\]
so \(\overline{L}=\underline{L}\). Let \(L=\overline{L}=\underline{L}\). Since
\[\underline{G}(r)\le G(r)\le \overline{G}(r),\]
it follows that \(\lim_{r\to b-} G(r)=L\).
We leave the proof of the following theorem to you (Exercise [exer:2]).
[theorem:3] Suppose \(g\) is locally integrable on \((a,b]\) and denote
\[G(r)=\int_{r}^{b}g(x)\,dx,\quad a\le r<b.\]
Then the improper integral \(\int_{a}^{b}g(x)\,dx\) converges if and only if\(,\) for each \(\epsilon >0,\) there is an \(r_{0}\in(a,b]\) such that
\[|G(r)-G(r_{1})|<\epsilon,\quad a<r,r_{1}\le r_{0}.\]
To see why we associate Theorems [theorem:2] and [theorem:3] with Cauchy, compare them with Theorem 4.3.5 (p. 204)