queries.jsonl

{"_id": "3", "text": "The rational numbers are dense in the reals$\\,;$ that is, if $a$ and $b$ are real numbers with $a<b,$ there is a rational number $p/q$ such that $a<p/q<b$."}
{"_id": "4", "text": "The set of irrational numbers is dense in the reals$\\,;$ that is, if $a$ and $b$ are real numbers with $a<b,$  there is an irrational number $t$ such that $a<t<b.$"}
{"_id": "7", "text": "Let $n_0$ be any integer $($positive$,$  negative$,$ or zero$)$$.$ Let $P_{n_0},$ $P_{n_0+1},$ \\dots$,$ $P_n,$ \\dots\\ be propositions$,$  one for each integer $n\\ge n_0,$ such that \\begin{alist} \\item % (a)  $P_{n_0}$ is true$\\,;$ \\item % (b)  for each integer $n\\ge n_0,$ $P_n$  implies $P_{n+1}.$ \\end{alist} Then $P_n$ is true for every integer  $n\\ge n_0.$"}
{"_id": "8", "text": "Let $n_0$ be any integer $($positive$,$  negative$,$ or zero$)$$.$ Let $P_{n_0},$ $P_{n_0+1}, $\\dots$,$ $P_n,$ \\dots\\ be propositions$,$  one for each integer $n\\ge n_0,$ such that \\begin{alist} \\item % (a)  $P_{n_0}$ is true$\\,;$ \\item % (b) for $n\\ge n_0,$ $P_{n+1}$ is true if $P_{n_0},$ $P_{n_0+1}, $\\dots$,$ $P_n$ are all true. \\end{alist} Then $P_n$ is true for $n\\ge n_0.$"}
{"_id": "11", "text": "If ${\\mathcal H}$ is an open covering of a closed and bounded subset $S$ of the real line$,$ then $S$ has an open covering $\\widetilde{\\mathcal H}$ consisting of finitely many open sets belonging to ${\\mathcal H}.$"}
{"_id": "12", "text": "Every bounded infinite set of real numbers has at least one limit point$.$"}
{"_id": "13", "text": "then it is unique$\\,;$ that is$,$ if \\begin{equation} \\label{eq:2.1.7} \\lim_{x\\to x_0} f(x)=L_1\\mbox{\\quad and \\quad}\\lim_{x\\to x_0} f(x)= L_2, \\end{equation} then $L_1=L_2.$"}
{"_id": "14", "text": "\\begin{equation}\\label{eq:2.1.9} \\lim_{x\\to x_0} f(x)=L_1\\mbox{\\quad and \\quad}\\lim_{x\\to x_0} g(x)= L_2, \\end{equation} then \\begin{eqnarray} \\lim_{x\\to x_0} (f+g)(x)\\ar= L_1+L_2,\\label{eq:2.1.10}\\\\ \\lim_{x\\to x_0} (f-g)(x)\\ar= L_1-L_2,\\label{eq:2.1.11}\\\\ \\lim_{x\\to x_0} (fg)(x)\\ar= L_1L_2,\\label{eq:2.1.12}\\\\ \\arraytext{and, if $L_2\\ne0$,}\\\\ \\lim_{x\\to x_0}\\left(\\frac{f}{g}\\right)(x)\\ar= \\frac{L_1}{ L_2}.\\label{eq:2.1.13} \\end{eqnarray}"}
{"_id": "17", "text": "If $f$ is bounded on $[a,x_0),$ then $\\beta=\\limsup_{x\\to x_0-}f(x)$ exists and is the unique real number with the following properties$\\,:$ \\begin{alist} \\item % (a) If $\\epsilon>0$, there is an $a_1$ in $[a,x_0)$ such that \\begin{equation} \\label{eq:2.1.22} f(x)<\\beta+\\epsilon\\mbox{\\quad if \\quad} a_1\\le x<x_0. \\end{equation} \\item % (b) If $\\epsilon>0$ and $a_1$ is in $[a,x_0),$ then $$ f(\\overline x)>\\beta-\\epsilon\\mbox{\\quad for some }\\overline x\\in[a_1,x_0). $$ \\end{alist}"}
{"_id": "22", "text": "If $f$ is continuous on a finite closed interval $[a,b],$ then $f$ is bounded on $[a,b].$"}
{"_id": "26", "text": "If $f$ is monotonic and nonconstant on $[a,b],$ then $f$ is continuous on $[a,b]$ if and only if its range $R_f=\\set{f(x)}{x\\in[a,b]}$ is the closed interval with endpoints $f(a)$ and $f(b).$"}
{"_id": "27", "text": "Suppose that $f$ is increasing and continuous on $[a,b],$ and let $f(a)=c$ and $f(b)=d.$ Then there is a unique function $g$ defined on $[c,d]$ such that \\begin{equation}\\label{eq:2.2.17} g(f(x))=x,\\quad a\\le x\\le b, \\end{equation} and \\begin{equation}\\label{eq:2.2.18} f(g(y))=y,\\quad c\\le y\\le d. \\end{equation} Moreover$,$ $g$ is continuous and increasing on $[c,d].$"}
{"_id": "29", "text": "If $f$ and $g$ are differentiable at $x_0,$ then so are $f+g,$ $f-g,$ and $fg,$ with \\begin{alist} \\item % (a)  $(f+g)'(x_0)=f'(x_0)+g'(x_0);$ \\item % (b)  $(f-g)'(x_0)=f'(x_0)-g(x_0);$ \\item % (c)  $(fg)'(x_0)=f'(x_0)g(x_0)+f(x_0)g'(x_0).$ \\end{alist}  The quotient $f/g$ is differentiable at $x_0$ if $g(x_0)\\ne0,$ with \\begin{alist} \\setcounter{lcal}{3} \\item % (d) $\\dst{\\left(\\frac{f}{g}\\right)' (x_0)= \\frac{f'(x_0)g(x_0)-f(x_0)g'(x_0)}{\\left[g(x_0)\\right]^2}}.$ \\end{alist}"}
{"_id": "30", "text": "The Chain Rule Suppose that  $g$ is differentiable at $x_0$ and $f$ is differentiable at $g(x_0).$ Then the composite function $h=f\\circ g,$ defined by $$ h(x)=f(g(x)), $$ is differentiable at $x_0,$ with $$ h'(x_0)=f'(g(x_0))g'(x_0). $$"}
{"_id": "31", "text": "If $f$ is differentiable at a local extreme point $x_0\\in D_{f}^{0},$ then $f'(x_0)=~0.$"}
{"_id": "32", "text": "Suppose that  $f$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b),$ and $f(a)=f(b).$ Then $f'(c)=0$ for some $c$ in the open interval $(a,b).$"}
{"_id": "33", "text": "Intermediate Value Theorem for Derivatives  Suppose that  $f$ is differentiable on $[a,b],$  $f'(a)\\ne f'(b),$ and $\\mu$ is between $f'(a)$ and $f'(b).$ Then $f'(c)=\\mu$ for some $c$ in $(a,b).$"}
{"_id": "35", "text": "Mean Value Theorem If $f$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b),$ then $$ f'(c)=\\frac{f(b)-f(a)}{ b-a} $$ for some $c$ in $(a,b).$"}
{"_id": "39", "text": "Suppose that $f$ and $g$ are differentiable and $g'$ has no zeros on $(a,b).$ Let \\begin{equation}\\label{eq:2.4.1} \\lim_{x\\to b-}f(x)=\\lim_{x\\to b-}g(x)=0 \\end{equation} \\newpage \\noindent or \\begin{equation}\\label{eq:2.4.2} \\lim_{x\\to b-}f(x)=\\pm\\infty\\mbox{\\quad and \\quad} \\lim_{x\\to b-}g(x)=\\pm\\infty, \\end{equation} and suppose that \\begin{equation}\\label{eq:2.4.3} \\lim_{x\\to b-}\\frac{f'(x)}{ g'(x)}=L\\quad\\mbox{$($finite or $\\pm \\infty)$}. \\end{equation} Then \\begin{equation}\\label{eq:2.4.4} \\lim_{x\\to b-}\\frac{f(x)}{ g(x)}=L. \\end{equation}"}
{"_id": "43", "text": "Extended Mean Value Theorem Suppose that $f$ is continuous on a finite closed interval $I$ with endpoints $a$ and $b$ $($that is, either $I=(a,b)$ or $I=(b,a)),$ $f^{(n+1)}$ exists on the open interval $I^0,$ and$,$ if $n>0,$ that $f'$, \\dots, $f^{(n)}$ exist and are continuous at $a.$ Then \\begin{equation}\\label{eq:2.5.17} f(b)-\\sum_{r=0}^n\\frac{f^{(r)}(a)}{ r!}(b-a)^r=\\frac{f^{(n+1)}(c)}{(n+1)!} (b-a)^{n+1} \\end{equation} for some $c$ in $I^0.$"}
{"_id": "44", "text": "If $f$ is unbounded on $[a,b],$ then $f$ is not integrable on $[a,b].$"}
{"_id": "45", "text": "Let $f$ be bounded on $[a,b]$,  and let $P$ be a partition of $[a,b].$  Then \\begin{alist} \\item % (a)  The upper sum $S(P)$ of $f$ over $P$ is the supremum  of the set of all Riemann sums of $f$ over $P.$ \\item % (b)  The lower sum $s(P)$ of $f$ over $P$ is the infimum  of the set of all Riemann sums of $f$ over $P.$ \\end{alist}"}
{"_id": "47", "text": "If $f$ is integrable on $[a,b],$ then $$ \\underline{\\int_a^b}f(x)\\,dx=\\overline{\\int_a^b}f(x)\\,dx=\\int_a^b f(x)\\,dx. $$"}
{"_id": "48", "text": "If $f$ is bounded on $[a,b]$ and \\begin{equation} \\label{eq:3.2.16} \\underline{\\int_a^b} f(x)\\,dx=\\overline{\\int_a^b}f(x)\\,dx=L, \\end{equation} then $f$ is integrable on $[a,b]$ and \\begin{equation} \\label{eq:3.2.17} \\int_a^b f(x)\\,dx=L. \\end{equation}"}
{"_id": "50", "text": "If $f$ is bounded on $[a,b],$ then $f$ is integrable on $[a,b]$  if and only if for each $\\epsilon>0$ there is a partition $P$ of $[a,b]$ for which \\begin{equation} \\label{eq:3.2.19} S(P)-s(P)<\\epsilon. \\end{equation}"}
{"_id": "51", "text": "If $f$ is continuous on $[a,b],$ then $f$ is integrable on $[a,b]$."}
{"_id": "52", "text": "If $f$ is monotonic on $[a,b],$ then $f$ is integrable on $[a,b]$."}
{"_id": "53", "text": "If $f$ and $g$ are integrable on $[a,b],$ then so is $f+g,$ and \\vskip4pt $$ \\int_a^b (f+g)(x)\\,dx=\\int_a^b f(x)\\,dx+\\int_a^b g(x)\\,dx. $$"}
{"_id": "56", "text": "If $f$ and $g$ are integrable on $[a,b]$ and $f(x)\\le g(x)$ for $a\\le x\\le b,$ then \\begin{equation}\\label{eq:3.3.1} \\int_a^b f(x)\\,dx\\le\\int_a^b g(x)\\,dx. \\end{equation}"}
{"_id": "57", "text": "If $f$ is integrable on $[a,b],$ then so is $|f|$, and \\begin{equation} \\label{eq:3.3.3} \\left|\\int_a^b f(x)\\,dx\\right|\\le\\int_a^b |f(x)|\\,dx. \\end{equation}"}
{"_id": "58", "text": "If $f$ and $g$ are integrable on $[a,b],$ then so is the product $fg.$"}
{"_id": "59", "text": "First Mean Value Theorem for Integrals Suppose that $u$ is continuous and $v$ is integrable and nonnegative on $[a,b].$ Then \\begin{equation} \\label{eq:3.3.8} \\int_a^b u(x)v(x)\\,dx=u(c)\\int_a^b v(x)\\,dx \\end{equation} for some $c$ in $[a,b]$."}
{"_id": "60", "text": "If $f$ is integrable on $[a,b]$ and $a\\le a_1<b_1\\le b,$ then $f$ is integrable on $[a_1,b_1].$"}
{"_id": "62", "text": "If $f$ is integrable on $[a,b]$ and $a\\le c\\le b,$ then the function $F$ defined by $$  F(x)=\\int_c^x f(t)\\,dt $$  satisfies a Lipschitz condition on $[a,b],$ and is therefore continuous on $[a,b].$"}
{"_id": "63", "text": "If $f$ is integrable on $[a,b]$ and $a\\le c\\le b,$ then $F(x)=\\int_c^x f(t)\\,dt$ is differentiable at any point $x_0$ in $(a,b)$ where $f$ is continuous$,$ with $F'(x_0)=f(x_0).$ If $f$ is continuous from the right at $a,$ then $F_+'(a)=f(a)$. If $f$ is continuous from the left at $b,$ then $F_-'(b)=f(b).$"}
{"_id": "64", "text": "Suppose that $F$ is continuous on the closed interval $[a,b]$ and differentiable on the open interval $(a,b),$ and $f$ is integrable on $[a,b].$ Suppose  also that $$ F'(x)=f(x),\\quad a<x<b. $$ Then \\begin{equation}\\label{eq:3.3.14} \\int_a^b f(x)\\,dx=F(b)-F(a). \\end{equation}"}
{"_id": "65", "text": "Fundamental Theorem of  Calculus If $f$ is continuous on $[a,b],$ then $f$ has an antiderivative on $[a,b].$ Moreover$,$ if $F$ is any antiderivative of $f$ on $[a,b],$ then $$ \\int_a^b f(x)\\,dx=F(b)-F(a). $$"}
{"_id": "66", "text": "Integration by Parts If $u'$ and $v'$ are integrable on $[a,b],$ then \\begin{equation}\\label{eq:3.3.16} \\int_a^b u(x)v'(x)\\,dx=u(x)v(x)\\bigg|^b_a-\\int_a^b v(x)u'(x)\\,dx. \\end{equation}"}
{"_id": "67", "text": "Second Mean Value Theorem for Integrals Suppose that $f'$ is nonnegative and integrable and $g$ is continuous on $[a,b].$  Then \\begin{equation}\\label{eq:3.3.17} \\int_a^b f(x)g(x)\\,dx=f(a)\\int_a^c g(x)\\,dx+f(b)\\int_c^b g(x)\\,dx \\end{equation} for some $c$ in $[a,b].$"}
{"_id": "68", "text": "Suppose that the transformation $x=\\phi(t)$ maps the interval $c\\le t\\le d$ into the interval $a\\le x\\le b,$ with $\\phi(c)=\\alpha$ and $\\phi(d)=\\beta,$ and let $f$ be continuous on $[a,b].$ Let $\\phi'$ be integrable on $[c,d].$ Then \\begin{equation}\\label{eq:3.3.21} \\int_\\alpha^\\beta f(x)\\,dx=\\int_c^d f(\\phi(t))\\phi'(t)\\,dt. \\end{equation}"}
{"_id": "69", "text": "Suppose that $\\phi'$ is integrable and $\\phi$ is monotonic on $[c,d],$ and the transformation $x=\\phi(t)$ maps $[c,d]$ onto $[a,b].$ Let $f$ be bounded on $[a,b].$ Then $$ g(t)=f(\\phi(t))\\phi'(t) $$ is integrable on $[c,d]$ if and only if $f$ is integrable over $[a,b],$ and in this case $$ \\int_a^b f(x)\\,dx=\\int_c^d f(\\phi(t))|\\phi'(t)|\\,dt. $$"}
{"_id": "70", "text": "Suppose that   $f_1,$ $f_2,$ \\dots$,$ $f_n$  are locally integrable on $[a,b)$ and that $\\int_a^bf_1(x)\\,dx,$ $\\int_a^bf_2(x)\\,dx,$ \\dots$,$ $\\int_a^bf_n(x)\\,dx$ converge$.$ Let $c_1,$ $c_2,$ \\dots$,$ $c_n$ be constants$.$ Then  $\\int_a^b(c_1f+c_2f_1+\\cdots+c_nf_n)(x)\\,dx$ converges and \\begin{eqnarray*} \\int_a^b (c_1f_1+c_2f_2+\\cdots+c_nf_n)(x)\\,dx\\ar=c_1\\int_a^b f_1(x)\\,dx +c_2\\int_a^b f_2(x)\\,dx\\\\ \\ar{}+\\cdots+c_n\\int_a^b f_n(x)\\,dx. \\end{eqnarray*}"}
{"_id": "71", "text": "If $f$ is nonnegative and locally integrable on $[a,b),$ then $\\int_a^b f(x)\\,dx$ converges if the function $$ F(x)=\\int_a^x f(t)\\,dt $$ is bounded on $[a,b)$, and $\\int_a^b f(x)\\,dx=\\infty$ if it is not. These are the only possibilities, and $$ \\int_a^b f(t)\\,dt=\\sup_{a\\le x<b}F(x) $$ in either case$.$"}
{"_id": "72", "text": "Comparison Test If $f$ and $g$ are locally integrable on $[a,b)$ and \\begin{equation}\\label{eq:3.4.2} 0\\le f(x)\\le g(x),\\quad a\\le x<b, \\end{equation} then \\vskip3pt \\noindent \\part{a}\\phantom{xxxxxxxxxxxxxxxxxxxx} $\\dst\\int_a^b f(x)\\,dx<\\infty \\mbox{\\quad if\\quad}\\dst\\int_a^b g(x)\\,dx<\\infty$ \\\\ \\vskip3pt \\noindent and\\\\ \\vskip3pt \\noindent \\part{b}\\phantom{xxxxxxxxxxxxxxxxxxxx} $\\dst\\int_a^b g(x)\\,dx= \\infty\\mbox{\\quad if\\quad}\\dst\\int_a^b f(x)\\,dx=\\infty$."}
{"_id": "73", "text": "Suppose that  $f$ and $g$ are locally integrable on $[a,b),$ $g(x)>0$ and $f(x)\\ge0$ on some subinterval $[a_1,b)$ of $[a,b),$ and \\begin{equation}\\label{eq:3.4.3} \\lim_{x\\to b-}\\frac{f(x)}{ g(x)}=M. \\end{equation} \\begin{alist} \\item % (a) If $0<M<\\infty,$ then $\\int_a^b f(x)\\,dx$ and $\\int_a^b g(x)\\,dx$ converge or diverge together. \\item % (b)  If $M=\\infty$ and $\\int_a^b g(x)\\,dx=\\infty,$ then $\\int_a^b f(x)\\,dx=\\infty$. \\item % (c)  If $M=0$ and $\\int_a^b g(x)\\,dx<\\infty,$ then $\\int_a^b f(x)\\,dx<\\infty$. \\end{alist}"}
{"_id": "74", "text": "If $f$ is locally integrable on $[a,b)$ and $\\int_a^b |f(x)|\\,dx<\\infty,$ then $\\int_a^b f(x)\\,dx$ converges$;$ that is$,$ an absolutely convergent integral is convergent$.$"}
{"_id": "75", "text": "Suppose that  $f$ is continuous  and its antiderivative $F(x)=\\int_a^x f(t)\\,dt$ is bounded on $[a,b).$ Let $g'$ be absolutely integrable on $[a,b),$ and   suppose that \\begin{equation}\\label{eq:3.4.9} \\lim_{x\\to b-} g(x)=0. \\end{equation} Then $\\int_a^b f(x)g(x)\\,dx$ converges$.$"}
{"_id": "76", "text": "Suppose that  $u$ is continuous on $[a,b)$ and $\\int_a^bu(x)\\,dx$   diverges$.$ Let $v$ be positive and differentiable on $[a,b),$ and  suppose that  $\\lim_{x\\to b-}v(x)=\\infty$ and $v'/v^2$ is absolutely integrable on $[a,b).$ Then $\\int_a^b u(x)v(x)\\,dx$  diverges$.$"}
{"_id": "80", "text": "A bounded function $f$ is integrable on a finite interval $[a,b]$ if and only if the set $S$ of discontinuities of $f$ in $[a,b]$ is of Lebesgue measure zero$.$"}
{"_id": "81", "text": "The limit of a convergent sequence is unique$.$"}
{"_id": "83", "text": "\\begin{alist} \\item % (a)  If $\\{s_n\\}$ is nondecreasing$,$ then $\\lim_{n\\to\\infty}s_n=\\sup\\{s_n\\}.$ \\item % (b If $\\{s_n\\}$ is nonincreasing$,$ then $\\lim_{n\\to\\infty}s_n= \\inf\\{s_n\\}.$ \\end{alist}"}
{"_id": "85", "text": "Let \\begin{equation}\\label{eq:4.1.4} \\lim_{n\\to\\infty} s_n=s\\mbox{\\quad and\\quad}\\lim_{n\\to\\infty} t_n=t, \\end{equation} where $s$ and $t$ are finite$.$  Then \\begin{equation}\\label{eq:4.1.5} \\lim_{n\\to\\infty} (cs_n)=cs \\end{equation} if $c$ is a constant$;$ \\begin{eqnarray} \\lim_{n\\to\\infty}(s_n+t_n)\\ar=s+t,\\label{eq:4.1.6}\\\\ \\lim_{n\\to\\infty}(s_n-t_n)\\ar=s-t, \\label{eq:4.1.7}\\\\ \\lim_{n\\to\\infty}(s_nt_n)\\ar=st,\\label{eq:4.1.8}\\\\ \\arraytext{and}\\nonumber\\\\ \\lim_{n\\to\\infty}\\frac{s_n}{ t_n}\\ar=\\frac{s}{ t}\\label{eq:4.1.9} \\end{eqnarray} if $t_n$ is nonzero for all $n$ and $t\\ne0$."}
{"_id": "86", "text": "\\begin{alist} \\item % (a) If $\\{s_n\\}$ is bounded above and does not diverge to $-\\infty,$ then there is a unique real number $\\overline{s}$ such that$,$ if $\\epsilon>0,$ \\begin{equation}\\label{eq:4.1.16} s_n<\\overline{s}+\\epsilon\\mbox{\\quad for large $n$} \\end{equation} and \\begin{equation}\\label{eq:4.1.17} s_n>\\overline{s}-\\epsilon\\mbox{\\quad for infinitely many  $n$}. \\end{equation} \\item % (b) If $\\{s_n\\}$ is bounded below and does not diverge to $\\infty,$ then there is a unique real number $\\underline{s}$ such that$,$ if $\\epsilon >0,$ \\begin{equation}\\label{eq:4.1.18} s_n>\\underline{s}-\\epsilon\\mbox{\\quad for large $n$} \\end{equation} and \\begin{equation}\\label{eq:4.1.19} s_n<\\underline{s}+\\epsilon\\mbox{\\quad for infinitely many $n$}. \\end{equation} \\end{alist}"}
{"_id": "87", "text": "Every sequence $\\{s_n\\}$ of real numbers has a unique limit superior$,$ $\\overline{s},$ and a unique limit inferior$,$ $\\underline{s}$, in the extended reals$,$ and \\begin{equation}\\label{eq:4.1.21} \\underline{s}\\le \\overline{s}. \\end{equation}"}
{"_id": "88", "text": "If $\\{s_n\\}$ is a sequence of real numbers, then \\begin{equation}\\label{eq:4.1.22} \\lim_{n\\to\\infty} s_n=s \\end{equation} if and only if \\begin{equation}\\label{eq:4.1.23} \\limsup_{n\\to\\infty}s_n=\\liminf_{n\\to\\infty} s_n=s. \\end{equation}"}
{"_id": "89", "text": "A sequence $\\{s_n\\}$ of real numbers converges if and only if$,$ for every  $\\epsilon>0,$ there is an integer $N$ such that \\begin{equation}\\label{eq:4.1.24} |s_n-s_m|<\\epsilon\\mbox{\\quad if\\quad} m,n\\ge N. \\end{equation}"}
{"_id": "91", "text": "If $\\{s_n\\}$ is monotonic and has a subsequence $\\{s_{n_k}\\}$ such that $$ \\lim_{k\\to\\infty} s_{n_k}=s\\quad (-\\infty\\le s\\le\\infty), $$ then $$ \\lim_{n\\to\\infty} s_n=s. $$"}
{"_id": "93", "text": "\\vspace*{3pt} \\begin{alist} \\item % (a)   If $\\{x_n\\}$ is bounded$,$ then $\\{x_n\\}$ has a convergent subsequence$.$ \\vspace*{3pt} \\item % (b)   If $\\{x_n\\}$ is unbounded above$,$  then $\\{x_n\\}$ has a subsequence $\\{x_{n_k}\\}$ such that $$ \\lim_{k\\to\\infty} x_{n_k}=\\infty. $$ \\vspace*{3pt} \\item % (c)   If $\\{x_n\\}$ is unbounded below$,$ then $\\{x_n\\}$ has a subsequence $\\{x_{n_k}\\}$ such that $$ \\lim_{k\\to\\infty} x_{n_k}=-\\infty. $$ \\end{alist}"}
{"_id": "95", "text": "If $f$ is continuous on a closed interval $[a,b],$ then $f$ is bounded on $[a,b].$"}
{"_id": "98", "text": "Cauchy's Convergence Criterion for Series A series $\\sum a_n$ converges if and only if for every $\\epsilon>0$ there is an integer $N$ such that \\begin{equation}\\label{eq:4.3.3} |a_n+a_{n+1}+\\cdots+a_m|<\\epsilon\\mbox{\\quad if\\quad} m\\ge n\\ge N. \\end{equation}"}
{"_id": "99", "text": "If $a_n\\ge0$ for $n\\ge k,$ then $\\sum a_n$ converges if its partial sums are bounded$,$ or diverges to $\\infty$ if they are not$.$ These are the only possibilities and$,$ in either case$,$ $$ \\sum_{n=k}^\\infty a_n =\\,\\sup\\set{A_n}{n\\ge k}\\negthickspace, $$ where $$ A_n=a_k+a_{k+1}+\\cdots+a_n,\\quad n\\ge k. $$"}
{"_id": "100", "text": "The Comparison Test Suppose that \\begin{equation}\\label{eq:4.3.5} 0\\le a_n\\le b_n,\\quad n\\ge k. \\end{equation} Then \\begin{alist} \\item % (a)  $\\sum a_n<\\infty$ if  $\\sum b_n<\\infty$$.$ \\item % (b)  $\\sum b_n=\\infty$  if  $\\sum a_n=\\infty.$ \\end{alist}"}
{"_id": "101", "text": "The Integral Test Let \\begin{equation}\\label{eq:4.3.7} c_n=f(n),\\quad n\\ge k, \\end{equation} where $f$ is  positive$,$ nonincreasing$,$ and locally integrable on $[k,\\infty).$ Then \\begin{equation}\\label{eq:4.3.8} \\sum c_n<\\infty \\end{equation} if and only if \\begin{equation}\\label{eq:4.3.9} \\int^\\infty_k f(x)\\,dx<\\infty. \\end{equation}"}
{"_id": "102", "text": "Suppose that  $a_n\\ge0$ and $b_n>0$ for $n\\ge k.$ Then \\begin{alist} \\item % (a) $\\dst{\\sum a_n<\\infty\\mbox{\\quad if\\quad}\\sum b_n< \\infty\\mbox{\\quad and\\quad}\\limsup_{n\\to\\infty} a_n/b_n<\\infty}.$ \\item % (b)  $\\dst{\\sum a_n=\\infty\\mbox{\\quad if\\quad}\\sum b_n= \\infty\\mbox{\\quad and\\quad}\\liminf_{n\\to\\infty} a_n/b_n>0}.$ \\end{alist}"}
{"_id": "103", "text": "Suppose that  $a_n>0,$ $b_n>0,$ and \\begin{equation}\\label{eq:4.3.12} \\frac{a_{n+1}}{ a_n}\\le \\frac{b_{n+1}}{ b_n}. \\end{equation} Then \\begin{alist} \\item % (a)  $\\sum a_n<\\infty$ if  $\\sum b_n<\\infty.$ \\item % (b)  $\\sum b_n=\\infty$  if  $\\sum a_n=\\infty.$ \\end{alist}"}
{"_id": "104", "text": "The Ratio Test Suppose that  $a_n>0$ for $n\\ge k.$  Then \\vspace*{5pt} \\begin{alist} \\vspace*{5pt} \\item % (a) $\\sum a_n<\\infty$ if\\, $\\limsup_{n\\to\\infty} a_{n+1}/a_n<1.$ \\vspace*{5pt} \\item % (b)  $\\sum a_n=\\infty$  if\\, $\\liminf_{n\\to\\infty} a_{n+1}/a_n>1.$ \\end{alist} \\vspace*{5pt} \\noindent If \\begin{equation}\\label{eq:4.3.13} \\liminf_{n\\to\\infty}\\frac{a_{n+1}}{ a_n}\\le1\\le \\limsup_{n\\to\\infty}\\frac{a_{n+1}}{ a_n}, \\end{equation} then the test is inconclusive$;$ that is$,$ $\\sum a_n$ may converge or diverge$.$"}
{"_id": "105", "text": "Suppose that  $a_n>0$ for large $n.$  Let $$ M=\\limsup_{n\\to\\infty} n\\left(\\frac{a_{n+1}}{ a_n}- 1\\right)\\mbox{\\quad and\\quad} m=\\liminf_{n\\to\\infty} n \\left(\\frac{a_{n+1}}{ a_n}-1\\right). $$ Then \\begin{alist} \\item % (a)  $\\sum a_n<\\infty$ if $M<-1.$ \\item % (b)  $\\sum a_n=\\infty$  if $m>-1.$ \\end{alist} The test is inconclusive if $m\\le-1\\le M.$"}
{"_id": "106", "text": "Cauchy's Root Test If $a_n\\ge 0$ for $n\\ge k,$ then \\begin{alist} \\item % (a)  $\\sum a_n<\\infty$ if $\\limsup_{n\\to\\infty} a^{1/n}_n<1.$ \\item % (b)  $\\sum a_n=\\infty$ if $\\limsup_{n\\to\\infty} a^{1/n}_n>1.$ \\end{alist} The test is inconclusive if $\\limsup_{n\\to\\infty} a^{1/n}_n= 1.$"}
{"_id": "108", "text": "Dirichlet's Test for Series The series $\\sum ^\\infty_{n=k} a_nb_n$ converges if $\\lim_{n\\to\\infty} a_n= 0,$ \\begin{equation}\\label{eq:4.3.18} \\sum |a_{n+1}-a_n|<\\infty, \\end{equation} and \\begin{equation}\\label{eq:4.3.19} |b_k+b_{k+1}+\\cdots+b_n|\\le M,\\quad n\\ge k, \\end{equation} for some constant $M.$"}
{"_id": "109", "text": "Suppose that  $\\sum_{n=k}^\\infty a_n=A,$ where $-\\infty \\le A\\le\\infty.$ Let $\\{n_j\\}_1^\\infty$ be an increasing sequence of integers, with $n_1\\ge k$. Define \\begin{eqnarray*} b_1\\ar=a_k+\\cdots+a_{n_1},\\\\ b_2\\ar=a_{{n_1}+1}+\\cdots+a_{n_2},\\\\ &\\vdots\\\\ b_r\\ar=a_{n_{r-1}+1}+\\cdots+a_{n_r}. \\end{eqnarray*} Then $$ \\sum_{j=1}^\\infty b_{n_j}=A. $$"}
{"_id": "110", "text": "If $\\sum_{n=1}^\\infty b_n$ is a rearrangement of an absolutely convergent series $\\sum_{n=1}^\\infty a_n,$ then $\\sum_{n=1}^\\infty b_n$ also converges absolutely$,$ and to the same sum$.$"}
{"_id": "112", "text": "Suppose that $\\sum_{n=1}^\\infty a_n$ is conditionally convergent and  $\\mu$ and $\\nu$ are arbitrarily given in  the extended reals$,$ with $\\mu\\le\\nu.$ Then the terms of $\\sum_{n=1}^\\infty a_n$ can be rearranged  to form a series $\\sum_{n=1}^\\infty b_n$ with partial sums $$ B_n=b_1+b_2+\\cdots+b_n,\\quad n\\ge1, $$ such that \\begin{equation}\\label{eq:4.3.25} \\limsup_{n\\to\\infty}B_n=\\nu\\mbox{\\quad and\\quad} \\liminf_{n\\to\\infty}B_n=\\mu. \\end{equation}"}
{"_id": "113", "text": "Let $$ \\sum_{n=0}^\\infty a_n=A\\mbox{\\quad and\\quad}\\sum_{n=0}^\\infty b_n=B, $$ where $A$ and $B$ are finite, and at least one term of each series is nonzero. Then $\\sum_{n=0}^\\infty p_n=AB$ for every sequence $\\{p_n\\}$ obtained by ordering the products in $\\eqref{eq:4.3.33}$ if and only if $\\sum a_n$ and $\\sum b_n$ converge absolutely$.$ Moreover$,$ in this case, $\\sum p_n$ converges absolutely$.$"}
{"_id": "114", "text": "If $\\sum_{n=0}^\\infty a_n$ and $\\sum_{n=0}^\\infty b_n$ converge absolutely to sums $A$ and $B,$ then the Cauchy product of $\\sum_{n=0}^\\infty a_n$ and $\\sum_{n=0}^\\infty b_n$ converges absolutely to $AB.$"}
{"_id": "117", "text": "Cauchy's Uniform Convergence Criterion A sequence of functions $\\{F_n\\}$ converges uniformly on a set $S$ if and only if for each $\\epsilon>0$ there is an integer $N$ such that \\begin{equation} \\label{eq:4.4.2} \\|F_n-F_m\\|_S<\\epsilon\\mbox{\\quad if\\quad} n, m\\ge N. \\end{equation}"}
{"_id": "121", "text": "Suppose that  $F'_n$ is continuous on $[a,b]$ for all $n\\ge1$ and $\\{F'_n\\}$ converges uniformly on $[a,b].$ Suppose also that  $\\{F_n(x_0)\\}$ converges for some $x_0$ in $[a,b].$ Then $\\{F_n\\}$ converges uniformly on $[a,b]$ to a differentiable limit function $F,$ and \\begin{equation} \\label{eq:4.4.11} F'(x)=\\lim_{n\\to\\infty}F'_n(x),\\quad a<x<b, \\end{equation} while \\begin{equation} \\label{eq:4.4.12} F'_+(a)=\\lim_{n\\to\\infty}F'_n(a+)\\mbox{\\quad and\\quad} F'_-(b)= \\lim_{n\\to\\infty}F'_n(b-). \\end{equation}"}
{"_id": "122", "text": "Cauchy's  Uniform Convergence Criterion A series $\\sum f_n$ converges uniformly on a set $S$ if and only if for each $\\epsilon>0$ there is an integer $N$ such that \\vskip0pt \\begin{equation} \\label{eq:4.4.16} \\|f_n+f_{n+1}+\\cdots+f_m\\|_S<\\epsilon\\mbox{\\quad if\\quad} m\\ge n\\ge N. \\end{equation}"}
{"_id": "123", "text": "Weierstrass's Test The series $\\sum f_n$ converges uniformly on $S$ if \\begin{equation} \\label{eq:4.4.17} \\|f_n\\|_S\\le M_n,\\quad n\\ge k, \\end{equation} where $\\sum M_n<\\infty.$"}
{"_id": "124", "text": "Dirichlet's Test for Uniform Convergence The series $$ \\sum_{n=k}^\\infty  f_ng_n $$  converges uniformly on $S$ if  $\\{f_n\\}$ converges uniformly to zero on $S,$  $\\sum (f_{n+1}-f_n)$ converges absolutely uniformly on $S,$ and \\begin{equation} \\label{eq:4.4.19} \\|g_k+g_{k+1}+\\cdots+g_n\\|_S\\le M,\\quad n\\ge k, \\end{equation} for some constant $M.$"}
{"_id": "128", "text": "In connection with the power series $\\eqref{eq:4.5.1},$ define $R$ in the extended reals by \\begin{equation}\\label{eq:4.5.2} \\frac{1}{ R}=\\limsup_{n\\to\\infty} |a_n|^{1/n}. \\end{equation} In particular$,$ $R=0$ if $\\limsup_{n\\to\\infty} |a_n|^{1/n}= \\infty$, and $R=\\infty$ if $\\limsup_{n\\to\\infty} |a_n|^{1/n}=0.$ Then the power series converges \\begin{alist} \\item % (a)  only for $x=x_0$ if $R=0;$ \\item % (b)  for all $x$ if $R=\\infty,$ and absolutely uniformly in every bounded set$;$ \\item % (c) for $x$ in $(x_0-R, x_0+R)$ if $0<R<\\infty,$ and absolutely uniformly in every closed subset of this interval. \\end{alist} The series diverges if $|x-x_0|>R.$ No general statement can be made concerning convergence at the endpoints $x=x_0+R$ and $x=x_0-R:$ the series may converge absolutely or conditionally at both$,$ converge conditionally at one and diverge at the other$,$ or diverge at both$.$"}
{"_id": "129", "text": "The radius of convergence of $\\sum a_n(x-x_0)^n$ is given by $$ \\frac{1}{ R}=\\lim_{n\\to\\infty}\\left|\\frac{a_{n+1}}{a_n}\\right| $$ if the limit exists in the extended reals$.$"}
{"_id": "130", "text": "A power series $$ f(x)=\\sum^\\infty_{n=0} a_n(x-x_0)^n $$ \\newpage \\noindent with positive radius of convergence $R$ is continuous and differentiable in its interval of convergence$,$ and its derivative can be obtained by differentiating term by term$;$ that is$,$ \\begin{equation} \\label{eq:4.5.9} f'(x)=\\sum^\\infty_{n=1} na_n(x-x_0)^{n-1}, \\end{equation} which can also be written as \\begin{equation} \\label{eq:4.5.10} f'(x)=\\sum^\\infty_{n=0}(n+1)a_{n+1} (x-x_0)^n. \\end{equation} This series also has radius of convergence $R.$"}
{"_id": "131", "text": "A power series $$ f(x)=\\sum_{n=0}^\\infty a_n(x-x_0)^n $$ with positive radius of convergence $R$  has derivatives of all orders in its interval of convergence$,$ which can be obtained by repeated term by term differentiation$;$ thus$,$ \\begin{equation}\\label{eq:4.5.11} f^{(k)}(x)=\\sum^\\infty_{n=k} n(n-1)\\cdots (n-k+1)a_n(x-x_0)^{n-k}. \\end{equation} The radius of convergence of each of these series is $R.$"}
{"_id": "135", "text": "If $f$ and $g$ are given by $\\eqref{eq:4.5.22}$ and $\\eqref{eq:4.5.23},$ then \\begin{eqnarray} f(x)g(x)\\ar=\\sum^\\infty_{n=0} c_n(x-x_0)^n,\\quad|x-x_0|<R,\\label{eq:4.5.24}\\\\ \\arraytext{where}\\nonumber c_n\\ar=\\sum^n_{r=0} a_rb_{n-r}=\\sum^n_{r=0} a_{n-r}b_r\\nonumber \\end{eqnarray} and $R\\ge\\min\\{R_1,R_2\\}.$"}
{"_id": "138", "text": "Triangle Inequality If $\\mathbf{X}$ and $\\mathbf{Y}$ are in $\\R^n,$ then \\begin{equation}\\label{eq:5.1.6} |\\mathbf{X}+\\mathbf{Y}|\\le |\\mathbf{X}|+|\\mathbf{Y}|, \\end{equation} with equality if and only if one of the vectors is a nonnegative multiple of the other$.$"}
{"_id": "142", "text": "Principle of Nested Sets If $S_1,$ $S_2,$ \\dots\\ are closed nonempty subsets of $\\R^n$ such that \\begin{equation}\\label{eq:5.1.14} S_1\\supset S_2\\supset\\cdots\\supset S_r\\supset\\cdots \\end{equation} and \\begin{equation}\\label{eq:5.1.15} \\lim_{r\\to\\infty} d(S_r)=0, \\end{equation} then the intersection $$ I=\\bigcap^\\infty_{r=1}S_r $$ contains exactly one point$.$"}
{"_id": "143", "text": "Heine--Borel Theorem If ${\\mathcal H}$ is an open covering of a compact subset $S,$  then $S$ can be covered by finitely many sets from ${\\mathcal H}.$"}
{"_id": "144", "text": "An open set $S$ in $\\R^n$ is connected if and only if it is polygonally connected$.$"}
{"_id": "153", "text": "Intermediate Value Theorem Let $f$ be continuous on a region $S$ in $\\R^n.$ Suppose that $\\mathbf{A}$ and $\\mathbf{B}$ are in $S$ and $$ f(\\mathbf{A})<u<f(\\mathbf{B}). $$ Then $f(\\mathbf{C})=u$ for some $\\mathbf{C}$ in $S.$"}
{"_id": "162", "text": "Suppose that  $f$ is defined in a neighborhood of $\\mathbf{X}_0$ in $\\R^n$  and $f_{x_1}(\\mathbf{X}_0),$ $f_{x_2}(\\mathbf{X}_{0}),$  \\dots$,$ $f_{x_n}(\\mathbf{X}_{0})$  exist$.$ Let $\\mathbf{X}_0$ be a local extreme point of $f.$ Then \\begin{equation}\\label{eq:5.3.42} f_{x_i}(\\mathbf{X}_0)=0,\\quad 1\\le i\\le n. \\end{equation}"}
{"_id": "163", "text": "The Chain Rule Suppose that  the real-valued function $f$ is differentiable at $\\mathbf{X}_0$ in $\\R^n,$ the vector-valued function $\\mathbf{G} =(g_1,g_2, \\dots,g_n)$ is differentiable at $\\mathbf{U}_0$ in $\\R^m,$ and $\\mathbf{X}_{0}  = \\mathbf{G}(\\mathbf{U}_0).$ Then the real-valued composite function $h=f\\circ\\mathbf{G}$ defined by \\begin{equation} \\label{eq:5.4.3} h(\\mathbf{U})=f(\\mathbf{G}(\\mathbf{U})) \\end{equation} is differentiable at $\\mathbf{U}_0,$ and \\begin{equation} \\label{eq:5.4.4} d_{\\mathbf{U}_0}h=f_{x_1}(\\mathbf{X}_0) d_{\\mathbf{U}_0}g_1+f_{x_2} (\\mathbf{X}_0) d_{\\mathbf{U}_0}g_2+\\cdots +f_{x_n} (\\mathbf{X}_0) d_{\\mathbf{U}_0}g_n. \\end{equation}"}
{"_id": "164", "text": "Mean Value Theorem for Functions of $\\mathbf n$ Variables Let $f$ be continuous at $\\mathbf{X}_1=(x_{11},x_{21}, \\dots, x_{n1})$ and $\\mathbf{X}_2=(x_{12},x_{22}, \\dots,x_{n2})$ and differentiable on the line segment $L$ from $\\mathbf{X}_1$ to $\\mathbf{X}_2.$ Then \\begin{equation} \\label{eq:5.4.21} f(\\mathbf{X}_2)-f(\\mathbf{X}_1)=\\sum_{i=1}^n f_{x_i} (\\mathbf{X}_0)(x_{i2}-x_{i1})=(d_{\\mathbf{X}_0}f)(\\mathbf{X}_2 -\\mathbf{X}_1) \\end{equation} for some $\\mathbf{X}_0$ on $L$ distinct from $\\mathbf{X}_1$ and $\\mathbf{X}_2$."}
{"_id": "165", "text": "Taylor's Theorem for Functions of $\\mathbf n$ Variables Suppose \\\\that  $f$ and its partial derivatives of order $\\le k$ are differentiable at $\\mathbf{X}_0$ and $\\mathbf{X}$ in $\\R^n$ and on the line segment $L$ connecting them$.$ Then \\begin{equation} \\label{eq:5.4.25} f(\\mathbf{X})=\\sum_{r=0}^k\\frac{1}{ r!} (d^{(r)}_{\\mathbf{X}_0}f)  (\\mathbf{X}-\\mathbf{X})+\\frac{1}{(k+1)!} (d^{(k+1)}_{\\widetilde{\\mathbf{X}}} f)(\\mathbf{X}-{\\mathbf{X}_0}) \\end{equation} for some $\\widetilde{\\mathbf{X}}$  on $L$ distinct from $\\mathbf{X}_0$ and $\\mathbf{X}$."}
{"_id": "166", "text": "Suppose that  $f$ and its partial derivatives of order $\\le k-1$ are differentiable in a neighborhood $N$ of a point $\\mathbf{X}_0$ in $\\R^n$ and all $k$th-order partial derivatives of $f$ are continuous at $\\mathbf{X}_0.$ Then \\begin{equation} \\label{eq:5.4.31} \\lim_{\\mathbf{X}\\to\\mathbf{X}_0} \\frac{f(\\mathbf{X})-T_k(\\mathbf{X})}{ |\\mathbf{X}-\\mathbf{X}_0|^k}=0. \\end{equation}"}
{"_id": "167", "text": "Suppose that  $f$ satisfies the hypotheses of Theorem~$\\ref{thmtype:5.4.9}$ with $k\\ge2,$ and  \\begin{equation} \\label{eq:5.4.38} d^{(r)}_{\\mathbf{X}_0} f\\equiv0\\quad (1\\le r\\le k-1),\\quad d^{(k)}_\\mathbf{X_0} f\\not\\equiv0. \\end{equation} Then \\begin{alist} \\item % (a) $\\mathbf{X}_0$ is not a local extreme point of $f$ unless $d^{(k)}_{\\mathbf{X}_0}f$ is semidefinite as a polynomial in $\\mathbf{X}-\\mathbf{X}_0.$ In particular$,$  $\\mathbf{X}_0$ is not a local extreme point of $f$ if $k$ is odd$.$ \\item % (b)  $\\mathbf{X}_0$ is a local minimum point of $f$ if $d^{(k)}_{\\mathbf{X}_0} f$ is positive definite$,$ or a local maximum point if $d^{(k)}_{\\mathbf{X}_0}f$ is negative definite$.$ \\item % (c)  If $d^{(k)}_{\\mathbf{X}_0}f$ is semidefinite$,$ then $\\mathbf{X}_0$ may be a local extreme point of $f,$ but it need not be$.$ \\end{alist}"}
{"_id": "175", "text": "Let $\\mathbf{A}$ be an $n\\times n$ matrix$.$ If $\\det(\\mathbf{A})=0,$ then $\\mathbf{A}$ is singular$.$ If $\\det(\\mathbf{A})\\ne0,$ then $\\mathbf{A}$ is nonsingular$,$ and $\\mathbf{A}$ has the unique inverse \\begin{equation} \\label{eq:6.1.10} \\mathbf{A}^{-1}=\\frac{1}{\\det(\\mathbf{A})}\\adj(\\mathbf{A}). \\end{equation}"}
{"_id": "176", "text": "The system $\\eqref{eq:6.1.11}$ has a solution $\\mathbf{X}$ for any given $\\mathbf{Y}$ if and only if $\\mathbf{A}$ is nonsingular$.$ In this case$,$ the solution is unique and is given by $\\mathbf{X}=\\mathbf{A}^{-1}\\mathbf{Y}$."}
{"_id": "177", "text": "If $\\mathbf{A}=[a_{ij}]$ is nonsingular$,$ then the solution of  the system \\begin{eqnarray*} a_{11}x_1+a_{12}x_2+\\cdots+a_{1n}x_n\\ar=y_1\\\\ a_{21}x_1+a_{22}x_2+\\cdots+a_{2n}x_n\\ar=y_2\\\\ &\\vdots& \\\\ a_{n1}x_1+a_{n2}x_2+\\cdots+a_{nn}x_n\\ar=y_n \\end{eqnarray*} $($or$,$ in matrix form$,$ $\\mathbf{AX}=\\mathbf{Y}$$)$ is given by $$ x_i=\\frac{D_i}{\\det(\\mathbf{A})},\\quad 1\\le i\\le n, $$ where $D_i$ is the determinant of the matrix obtained by replacing the $i$th column of $\\mathbf{A}$ with $\\mathbf{Y};$ thus$,$ $$ D_1=\\left|\\begin{array}{cccc} y_1&a_{12}&\\cdots&a_{1n}\\\\ y_2&a_{22}&\\dots&a_{2n}\\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\ y_n&a_{n2}&\\cdots&a_{nn}\\end{array}\\right|,\\quad D_2=\\left|\\begin{array}{ccccc} a_{11}&y_1&a_{13}&\\cdots&a_{1n}\\\\ a_{21}&y_2&a_{23}&\\cdots&a_{2n}\\\\ \\vdots&\\vdots&\\vdots&\\ddots&\\vdots\\\\ a_{n1}&y_n&a_{n3}&\\cdots&a_{nn}\\end{array}\\right|,\\quad\\cdots, $$ $$ D_n=\\left|\\begin{array}{cccc}  a_{11}&\\cdots&a_{1,n-1}&y_1\\\\ a_{21}&\\cdots&a_{2,n-1}&y_2\\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\ a_{n1}&\\cdots&a_{n,n-1}&y_n\\end{array}\\right|. $$"}
{"_id": "181", "text": "A transformation $\\mathbf{F}=(f_1,f_2, \\dots,f_m)$ defined in a neighborhood of $\\mathbf{X}_0\\in\\R^n$  is differentiable at $\\mathbf{X}_0$ if and only if there is a constant $m\\times n$ matrix $\\mathbf{A}$ such that \\begin{equation}\\label{eq:6.2.2} \\lim_{\\mathbf{X}\\to\\mathbf{X}_0} \\frac{ \\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{X}_0)-\\mathbf{A} (\\mathbf{X}-\\mathbf{X}_0)} {|\\mathbf{X}-\\mathbf{X}_0|}=\\mathbf{0}.  \\end{equation} If $\\eqref{eq:6.2.2}$ holds$,$ then $\\mathbf{A}$ is given uniquely by \\begin{equation}\\label{eq:6.2.3} \\mathbf{A}=\\left[\\frac{\\partial f_i(\\mathbf{X}_0)}{\\partial x_j}\\right]= \\left[\\begin{array}{cccc}\\dst{\\frac{\\partial f_1(\\mathbf{X}_0)}{\\partial x_1}}& \\dst{\\frac{\\partial f_1(\\mathbf{X}_0)}{\\partial x_2}}&\\cdots& \\dst{\\frac{\\partial f_1(\\mathbf{X}_0)}{\\partial x_n}}\\\\ [3\\jot] \\dst{\\frac{\\partial f_2(\\mathbf{X}_0)}{\\partial x_1}}& \\dst{\\frac{\\partial f_2(\\mathbf{X}_0)}{\\partial x_2}}& \\cdots&\\dst{\\frac{\\partial f_2(\\mathbf{X}_0)}{\\partial x_n}}\\\\ \\vdots&\\vdots&\\ddots&\\vdots\\\\ \\dst{\\frac{\\partial f_m(\\mathbf{X}_0)}{\\partial x_1}}& \\dst{\\frac{\\partial f_m(\\mathbf{X}_0)}{\\partial x _2}}& \\cdots&\\dst{\\frac{\\partial f_m(\\mathbf{X}_0)}{\\partial x_n}} \\end{array}\\right]. \\end{equation}"}
{"_id": "183", "text": "Let  $\\mathbf{F}=(f_1,f_2, \\dots,f_m):\\R^n\\to\\R^m,$ and suppose that the partial derivatives \\begin{equation}\\label{eq:6.2.7} \\frac{\\partial f_i}{\\partial x_j},\\quad 1\\le i\\le m,\\quad 1\\le j\\le n, \\end{equation} exist on a neighborhood of $\\mathbf{X}_0$ and are continuous at $\\mathbf{X}_0.$ Then $\\mathbf{F}$ is differentiable at $\\mathbf{X}_0.$"}
{"_id": "184", "text": "Suppose that $\\mathbf{F}: \\R^n\\to\\R^m$ is differentiable at $\\mathbf{X}_0,$ $\\mathbf{G}:\\R^k\\to\\R^n$ is differentiable at $\\mathbf{U}_0,$ and $\\mathbf{X}_0=\\mathbf{G}(\\mathbf{U}_0).$ Then the composite function $\\mathbf{H}=\\mathbf{F}\\circ\\mathbf{G}:\\R^k\\to\\R^m,$ defined by $$ \\mathbf{H}(\\mathbf{U})=\\mathbf{F}(\\mathbf{G}(\\mathbf{U})), $$ is differentiable at $\\mathbf{U}_0.$ Moreover$,$ \\begin{equation}\\label{eq:6.2.22} \\mathbf{H}'(\\mathbf{U}_0)=\\mathbf{F}'(\\mathbf{G}(\\mathbf{U}_0)) \\mathbf{G}'(\\mathbf{U}_0) \\end{equation} and \\begin{equation}\\label{eq:6.2.23} d_{\\mathbf{U}_0}\\mathbf{H}=d_{\\mathbf{X}_0}\\mathbf{F}\\circ d_{\\mathbf{U}_0}\\mathbf{G}, \\end{equation} where $\\circ$ denotes composition$.$"}
{"_id": "186", "text": "Suppose that $\\mathbf{F}: \\R^n\\to \\R^n$ is regular on an open set $S,$ and let $\\mathbf{G}=\\mathbf{F}^{-1}_S.$ Then $\\mathbf{F}(S)$ is open$,$ $\\mathbf{G}$ is continuously differentiable on $\\mathbf{F}(S),$ and $$ \\mathbf{G}'(\\mathbf{U})=(\\mathbf{F}'(\\mathbf{X}))^{-1}, \\mbox{\\quad where\\quad}\\mathbf{U}=\\mathbf{F}(\\mathbf{X}). $$ Moreover$,$ since $\\mathbf{G}$ is one-to-one on $\\mathbf{F}(S),$  $\\mathbf{G}$ is regular on $\\mathbf{F}(S).$"}
{"_id": "187", "text": "The Inverse Function Theorem Let $\\mathbf{F}: \\R^n\\to \\R^n$ be continuously differentiable on an open set $S,$ and suppose that $J\\mathbf{F}(\\mathbf{X})\\ne0$ on $S.$ Then$,$ if $\\mathbf{X}_0\\in S,$ there is an open neighborhood $N$ of $\\mathbf{X}_0$ on which $\\mathbf{F}$ is regular$.$ Moreover$,$ $\\mathbf{F}(N)$ is open and $\\mathbf{G}= \\mathbf{F}^{-1}_N$ is continuously differentiable on $\\mathbf{F}(N),$ with $$ \\mathbf{G}'(\\mathbf{U})=\\left[\\mathbf{F}'(\\mathbf{X})\\right]^{-1}\\quad \\mbox{ $($where $\\mathbf{U}=\\mathbf{F}(\\mathbf{X})$$)$},\\quad \\mathbf{U}\\in\\mathbf{F}(N). $$"}
{"_id": "188", "text": "The Implicit Function Theorem Suppose that  $\\mathbf{F}:\\R^{n+m}\\to \\R^m$ is continuously differentiable on an open set $S$ of $\\R^{n+m}$ containing $(\\mathbf{X}_0,\\mathbf{U}_0).$ Let $\\mathbf{F}(\\mathbf{X}_0,\\mathbf{U}_0)=\\mathbf{0},$ and suppose that  $\\mathbf{F}_\\mathbf{U}(\\mathbf{X}_0,\\mathbf{U}_0)$ is nonsingular$.$ Then there is a neighborhood $M$ of  $(\\mathbf{X}_0,\\mathbf{U}_{0}),$  contained in $S,$ on which  $\\mathbf{F}_\\mathbf{U}(\\mathbf{X},\\mathbf{U})$  is nonsingular  and a neighborhood $N$ of $\\mathbf{X}_0$ in $\\R^n$ on which a unique  continuously differentiable  transformation $\\mathbf{G}: \\R^n\\to \\R^m$ is defined$,$ such that $\\mathbf{G}(\\mathbf{X}_0)=\\mathbf{U}_0$ and \\begin{equation} \\label{eq:6.4.6} (\\mathbf{ X},\\mathbf{G}(\\mathbf{X}))\\in M\\mbox{\\quad and \\quad} \\mathbf{F}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))=0\\mbox{\\quad  if}\\quad\\mathbf{X}\\in N. \\end{equation} Moreover$,$ \\begin{equation} \\label{eq:6.4.7} \\mathbf{G}'(\\mathbf{X})=-[\\mathbf{F}_\\mathbf{U}(\\mathbf{X},\\mathbf{G}(\\mathbf{X}))]^{-1} \\mathbf{F}_\\mathbf{X}(\\mathbf{X},\\mathbf{G}(\\mathbf{X})),\\quad \\mathbf{X}\\in N. \\end{equation}"}
{"_id": "189", "text": "If $f$ is unbounded on the nondegenerate rectangle $R$ in $\\R^n,$ then $f$ is not integrable on $R.$"}
{"_id": "196", "text": "If $f$ is continuous on a rectangle $R$ in $\\R^n,$ then $f$ is integrable on~$R.$"}
{"_id": "197", "text": "Suppose that $f$ is bounded on a rectangle \\begin{equation}\\label{eq:7.1.30} R=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_n,b_n] \\end{equation} and continuous except on a subset $E$ of $R$ with zero content$.$ Then $f$ is integrable on $R.$"}
{"_id": "198", "text": "Suppose that $f$ is bounded on a bounded set $S$ and continuous except on a subset $E$ of $S$ with zero content. Suppose also that $\\partial S$ has zero content$.$ Then $f$ is integrable on $S.$"}
{"_id": "199", "text": "A differentiable surface in $\\R^n$ has zero content$.$"}
{"_id": "207", "text": "If $f$ is integrable on disjoint sets $S_1$ and $S_2,$ then $f$ is integrable on $S_1\\cup S_2,$ and \\begin{equation}\\label{eq:7.1.39} \\int_{S_1\\cup S_2} f(\\mathbf{X})\\,d\\mathbf{X}= \\int_{S_1} f(\\mathbf{X})\\,d\\mathbf{X}+ \\int_{S_2} f(\\mathbf{X})\\,d\\mathbf{X}. \\end{equation}"}
{"_id": "208", "text": "$R= [a,b]\\times [c,d]$ and $$  F(y)=\\int_a^b f(x,y)\\,dx $$ exists for each $y$ in $[c,d].$ Then $F$ is integrable on $[c,d],$ and \\begin{equation}\\label{eq:7.2.1} \\int_c^d F(y)\\,dy=\\int_R f(x,y)\\,d(x,y); \\end{equation} that is$,$ \\begin{equation}\\label{eq:7.2.2} \\int_c^d dy\\int_a^b f(x,y)\\,dx=\\int_R f(x,y)\\,d(x,y). \\end{equation}"}
{"_id": "209", "text": "Let $I_1,$ $I_2,$ \\dots$,$ $I_n$ be closed intervals and suppose that  $f$ is integrable on $R=I_1\\times I_2\\times\\cdots\\times I_n.$ Suppose that  there is an integer $p$ in $\\{1,2, \\dots,n-1\\}$ such that $$ F_p(x_{p+1},x_{p+2}, \\dots,x_n)=\\int_{I_1\\times I_2\\times\\cdots\\times I_p} f(x_1,x_2, \\dots,x_n)\\,d(x_1,x_2, \\dots,x_p) $$ exists for each $(x_{p+1},x_{p+2}, \\dots,x_n)$ in $I_{p+1}\\times I_{p+2}\\times\\cdots\\times I_n.$ Then $$ \\int_{I_{p+1}\\times I_{p+2}\\times\\cdots\\times I_n} F_p(x_{p+1}, x_{p+2}, \\dots,x_n)\\,d(x_{p+1},x_{p+2}, \\dots,x_n) $$ exists and equals $\\int_R f(\\mathbf{X})\\,d\\mathbf{X}$."}
{"_id": "210", "text": "Let $I_j=[a_j,b_j],$ $1\\le j\\le n$, and suppose that $f$ is integrable on $R=I_1\\times I_2 \\times\\cdots\\times I_n.$ Suppose also that  the integrals $$ F_p(x_{p+1}, \\dots,x_n)=\\int_{I_1\\times I_2\\cdots\\times I_p} f(\\mathbf{X}) \\,d(x_1,x_2, \\dots,x_p),\\quad1\\le p\\le n-1, $$ exist for all $$ (x_{p+1}, \\dots,x_n)\\mbox{\\quad in\\quad} I_{p+1}\\times\\cdots\\times I_n. $$ Then the iterated integral $$ \\int^{b_n}_{a_n} dx_n\\int^{b_{n-1}}_{a_{n-1}} dx_{n-1}\\cdots \\int^{b_2}_{a_2} dx_2\\int^{b_1}_{a_1} f(\\mathbf{X})\\,dx_1 $$ exists and equals $\\int_R f(\\mathbf{X})\\,d\\mathbf{X}.$"}
{"_id": "214", "text": "A bounded set $S$ is Jordan measurable if and only if the boundary of $S$ has zero content$.$"}
{"_id": "215", "text": "Suppose that  $\\mathbf{G}:\\R^n\\to \\R^n$ is regular on a compact Jordan measurable set $S.$ Then $\\mathbf{G}(S)$ is compact and Jordan measurable$.$"}
{"_id": "217", "text": "\\E^n\\to \\R^n$ is regular on a compact Jordan measurable set $S$ and $f$ is continuous on $\\mathbf{G}(S).$ Then \\begin{equation}\\label{eq:7.3.28} \\int_{\\mathbf{G}(S)} f(\\mathbf{X})\\,d\\mathbf{X}= \\int_S f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}. \\end{equation}"}
{"_id": "218", "text": "Suppose that  $\\mathbf{G}: \\E^n\\to \\R^n$ is continuously differentiable on a bounded open set $N$ containing the compact Jordan measurable set $S,$ and regular on $S^0.$ Suppose   also that $\\mathbf{G}(S)$ is Jordan measurable$,$ $f$ is continuous on $\\mathbf{G}(S),$ and $G(C)$ is Jordan measurable for every cube $C\\subset N$. Then \\begin{equation}\\label{eq:7.3.50} \\int_{\\mathbf{G}(S)} f(\\mathbf{X})\\,d\\mathbf{X}= \\int_S f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}. \\end{equation}"}
{"_id": "220", "text": "If $x$ and $y$ are vectors in  a normed vector space $(A,N),$ then \\begin{equation} \\label{eq:8.1.2} |N(x)-N(y)|\\le N(x-y). \\end{equation}"}
{"_id": "228", "text": "Any two norms  $N_1$ and $N_2$ on $\\R^n$ induce equivalent metrics on~$\\R^n.$"}
{"_id": "232", "text": "If  $T$ is  compact$,$  then every Cauchy sequence $\\{t_n\\}_{n=1}^\\infty$ in $T$ converges to a limit in $T.$"}
{"_id": "233", "text": "If $T$ is compact$,$ then $T$ is closed and bounded."}
{"_id": "234", "text": "If $T$  is compact$,$ then $T$ is totally bounded."}
{"_id": "235", "text": "If $(A,\\rho)$ is complete and $T$  is closed and totally bounded$,$ then $T$ is compact."}
{"_id": "236", "text": "A nonempty subset $T$ of $C[a,b]$ is compact if and only if it is closed$,$ uniformly bounded$,$ and equicontinuous."}
{"_id": "241", "text": "If $f$ is continuous on a compact set $T,$ then $f(T)$ is compact."}
{"_id": "242", "text": "If $f$ is continuous on a compact set $T,$ then $f$ is uniformly continuous on $T$."}
{"_id": "243", "text": "Contraction Mapping Theorem If $f$ is a contraction of a complete metric space $(A,\\rho),$ then the equation \\begin{equation} \\label{eq:8.3.8} f(u)=u \\end{equation} has a unique solution$.$"}
{"_id": "244", "text": "If $f$ is differentiable at $x_0,$ then \\begin{equation}\\label{eq:2.3.3} f(x)=f(x_0)+[f'(x_0)+E(x)](x-x_0), \\end{equation} where $E$ is defined on a neighborhood of $x_0$ and $$ \\lim_{x\\to x_0} E(x)=E(x_0)=0. $$"}
{"_id": "247", "text": "If $f$ is bounded on $[a,b]$ and  $\\epsilon>0,$   there is a $\\delta>0$ such that \\begin{equation} \\label{eq:3.2.12} \\overline{\\int_a^b}f(x)\\,dx\\le S(P)<\\overline{\\int_a^b}f(x)\\,dx+\\epsilon \\end{equation} and $$ \\underline{\\int_a^b} f(x)\\,dx\\ge s(P)>\\underline{\\int_a^b} f(x)\\,dx-\\epsilon $$ if $\\|P\\|<\\delta$."}
{"_id": "248", "text": "If $w_f(x)<\\epsilon$ for $a\\le x \\le b,$ then there is a $\\delta>0$ such that $W_f[a_1,b_1]\\le\\epsilon,$ provided that $[a_1,b_1]\\subset [a,b]$ and $b_1-a_1<\\delta.$"}
{"_id": "249", "text": "Let $f$ be bounded on $[a,b]$ and define $$ E_\\rho=\\set{x\\in [a,b]}{w_f(x)\\ge\\rho}. $$ Then $E_\\rho$ is closed$,$ and  $f$ is integrable on $[a,b]$ if and only if for every pair of positive numbers $\\rho$ and $\\delta,$ $E_\\rho$ can be covered by finitely many open intervals $I_1,$ $I_2, $\\dots$,$ $I_p$ such that \\begin{equation} \\label{eq:3.5.3} \\sum_{j=1}^p L(I_j)<\\delta. \\end{equation}"}
{"_id": "250", "text": "Suppose that for $n$ sufficiently large  $($that is$,$ for $n \\ge\\mbox{some integer }N$$)$  the terms of $\\sum_{n=k}^\\infty a_n$ satisfy  some condition that implies convergence of an infinite series$.$ Then $\\sum_{n=k}^\\infty a_n$ converges$.$ Similarly, suppose that for $n$  sufficiently large  the terms $\\sum_{n=k}^\\infty a_n$ satisfy  some condition that implies divergence of an infinite series$.$ Then $\\sum_{n=k}^\\infty a_n$ diverges$.$"}
{"_id": "255", "text": "Suppose that  $\\mathbf{G}=(g_1,g_2, \\dots,g_n)$ is differentiable at $$  \\mathbf{U}_0=(u_{10}, u_{20}, \\dots,u_{m0}), $$  and  define $$ M=\\left(\\sum_{i=1}^n\\sum_{j=1}^m\\left(\\frac{\\partial g_i(\\mathbf{U}_0} {\\partial u_j}\\right)^2\\right)^{1/2}. $$ Then$,$ if $\\epsilon>0,$ there is a $\\delta>0$ such that $$ \\frac{|\\mathbf{G}(\\mathbf{U})-\\mathbf{G}(\\mathbf{U}_0)|} {|\\mathbf{U}-\\mathbf{U}_{0}|} <M+\\epsilon\\mbox{\\quad if \\quad}0<|\\mathbf{U}-\\mathbf{U}_0|<\\delta. $$"}
{"_id": "256", "text": "Suppose that $\\mathbf{F}:\\R^n\\to\\R^m$ is continuously differentiable on a neighborhood $N$ of $\\mathbf{X}_0.$ Then$,$ for every $\\epsilon>0,$ there is a $\\delta>0$ such that \\begin{equation}\\label{eq:6.2.8} |\\mathbf{F}(\\mathbf{X})-\\mathbf{F}(\\mathbf{Y})|< (\\|\\mathbf{F}'(\\mathbf{X}_{0})\\| +\\epsilon) |\\mathbf{X}-\\mathbf{Y}| \\mbox{\\quad if\\quad}\\mathbf{A},\\mathbf{Y}\\in B_\\delta (\\mathbf{X}_0). \\end{equation}"}
{"_id": "258", "text": "If $\\mathbf{F}:\\R^n\\to\\R^m$ is continuously differentiable on an open set containing a compact set $D,$ then there is a constant $M$ such that \\begin{equation}\\label{eq:6.2.18} |\\mathbf{F}(\\mathbf{Y})-\\mathbf{F}(\\mathbf{X})|\\le M|\\mathbf{Y}-\\mathbf{X}| \\mbox{\\quad if\\quad}\\mathbf{X},\\mathbf{Y}\\in D. \\end{equation}"}
{"_id": "259", "text": "Suppose that $|f(\\mathbf{X})|\\le M$ if $\\mathbf{X}$ is in the rectangle $$ R=[a_1,b_1]\\times [a_2,b_2]\\times\\cdots\\times [a_n,b_n]. $$ Let ${\\bf P}=P_1\\times P_2\\times\\cdots\\times P_n$ and ${\\bf P}'= P_1'\\times P_2'\\times\\cdots\\times P_n'$ be partitions of $R,$ where $P_j'$ is obtained by adding $r_j$ partition points to $P_j,$ $1\\le j\\le n.$ Then \\begin{equation}\\label{eq:7.1.16} S({\\bf P})\\ge S({\\bf P}')\\ge S({\\bf P})-2MV(R)\\left(\\sum_{j=1}^n \\frac{r_j}{ b_j-a_j}\\right)\\|{\\bf P}\\| \\end{equation} and \\begin{equation}\\label{eq:7.1.17} s({\\bf P})\\le s({\\bf P}')\\le s({\\bf P})+2MV(R)\\left(\\sum_{j=1}^n \\frac{r_j }{ b_j-a_j}\\right)\\|{\\bf P}\\|. \\end{equation}"}
{"_id": "261", "text": "The union of finitely many sets with zero content has zero content$.$"}
{"_id": "262", "text": "Suppose that $S$ is contained in a bounded set $T$ and $f$ is integrable on $S.$ Then  $f_S$ $($see $\\eqref{eq:7.1.36})$ is integrable on $T,$ and $$ \\int_T f_S(\\mathbf{X})\\,d\\mathbf{X}=\\int_S f(\\mathbf{X})\\,d\\mathbf{X}. $$"}
{"_id": "264", "text": "Suppose that  $\\mathbf{G}: \\R^n\\to \\R^n$ is continuously differentiable on a bounded  open set $S,$ and let $K$ be a closed subset of $S$ with zero content$.$ Then $\\mathbf{G}(K)$ has zero content."}
{"_id": "265", "text": "A nonsingular $n\\times n$ matrix $\\mathbf{A}$ can be written as \\begin{equation}\\label{eq:7.3.10} \\mathbf{A}=\\mathbf{E}_k\\mathbf{E}_{k-1}\\cdots\\mathbf{E}_1, \\end{equation} where each $\\mathbf{E}_i$ is a matrix that can be obtained from the $n\\times n$ identity matrix $\\mathbf{I}$ by one of the following operations$:$ \\begin{alist} \\item % (a) interchanging two rows of $\\mathbf{I};$ \\item % (b) multiplying a row of $\\mathbf{I}$ by a nonzero constant$;$ \\item % (c) adding a multiple of one row of $\\mathbf{I}$ to another$.$ \\end{alist}"}
{"_id": "266", "text": "Suppose that  $\\mathbf{G}:\\E^n\\to \\R^n$ is regular  on a cube $C$ in $\\E^n,$ and let $\\mathbf{A}$ be a nonsingular $n\\times n$ matrix$.$ Then \\begin{equation}\\label{eq:7.3.29} V(\\mathbf{G}(C))\\le |\\det(\\mathbf{A})|\\left[\\max \\set{\\|\\mathbf{A}^{-1}\\mathbf{G}'(\\mathbf{Y})\\|_\\infty}{\\mathbf{Y}\\in C} \\right]^n V(C). \\end{equation}"}
{"_id": "267", "text": "If $\\mathbf{G}:\\E^n\\rightarrow \\R^n$  is regular on a cube $C$ in $\\R^n,$ then \\begin{equation}\\label{eq:7.3.32} V(\\mathbf{G}(C))\\le\\int_C |JG(\\mathbf{Y})|\\,d\\mathbf{Y}. \\end{equation}"}
{"_id": "269", "text": "Suppose that  $\\mathbf{G}: \\E^n\\to \\R^n$ is regular on a compact Jordan measurable set $S$ and $f$ is continuous and nonnegative on $\\mathbf{G}(S).$ Let \\begin{equation}\\label{eq:7.3.37} Q(S)=\\int_{\\mathbf{G}(S)} f(\\mathbf{X})\\,d\\mathbf{X}-\\int_S  f(\\mathbf{G}(\\mathbf{Y})) |J\\mathbf{G}(\\mathbf{Y})|\\,d\\mathbf{Y}. \\end{equation} Then $Q(S)\\le0.$"}
{"_id": "270", "text": "Under the assumptions of Lemma~$\\ref{thmtype:7.3.13},$ $Q(S)\\ge0.$"}
{"_id": "275", "text": "If $f$ is continuous on a set $T,$ then  $f$ is  uniformly continuous on any finite closed interval contained in $T.$"}
{"_id": "276", "text": "If $f'$ is integrable on $[a,b],$ then $$ \\int_a^b f'(x)\\,dx=f(b)-f(a). $$"}
{"_id": "278", "text": "If $\\sum a_n$ converges$,$ then for each $\\epsilon>0$ there is an integer $K$ such that $$ \\left|\\sum_{n=k}^\\infty a_n\\right|<\\epsilon\\mbox{\\quad if\\quad} k\\ge K; $$ that is$,$ $$ \\lim_{k\\to\\infty}\\sum_{n=k}^\\infty a_n=0. $$"}
{"_id": "279", "text": "Suppose that  $a_n\\ge0$ and $b_n>0$ for $n\\ge k,$ and $$ \\lim_{n\\to\\infty}\\frac{a_n}{ b_n}=L, $$ where $0<L<\\infty.$ Then $\\sum a_n$ and $\\sum b_n$ converge or diverge together$.$"}
{"_id": "280", "text": "Suppose that  $a_n>0\\ (n\\ge k)$ and $$ \\lim_{n\\to\\infty}\\frac{a_{n+1}}{ a_n}=L. $$ \\vskip-1em Then \\begin{alist} \\item % (a)  $\\sum a_n<\\infty$ if  $L<1.$ \\item % (b)  $\\sum a_n=\\infty$  if  $L>1.$ \\end{alist} The test is inconclusive if $L=1.$"}
{"_id": "284", "text": "If $\\sum f_n$ converges uniformly on $S,$ then $\\lim_{n\\to\\infty}\\|f_n\\|_S=0.$"}
{"_id": "285", "text": "The series $\\sum_{n=k}^\\infty  f_ng_n$ converges uniformly on $S$ if $$ f_{n+1}(x)\\le f_n(x),\\quad x\\in S,\\quad n\\ge k, $$ $\\{f_n\\}$ converges uniformly to zero on $S,$ and $$ \\|g_k+g_{k+1}+\\cdots+g_n\\|_S\\le M,\\quad n\\ge k, $$ for some constant $M.$"}
{"_id": "287", "text": "Uniqueness of Power Series If \\begin{equation}\\label{eq:4.5.13} \\sum^\\infty_{n=0} a_n(x-x_0)^n=\\sum^\\infty_{n=0} b_n(x-x_0)^n \\end{equation} for all $x$ in some interval $(x_0-r,x_0+r),$ then \\begin{equation}\\label{eq:4.5.14} a_n=b_n,\\quad n\\ge0. \\end{equation}"}
{"_id": "288", "text": "If $\\mathbf{X},$ $\\mathbf{Y},$ and $\\mathbf{Z}$ are  in $\\R^n,$ then $$ |\\mathbf{X}-\\mathbf{Z}|\\le |\\mathbf{X}-\\mathbf{Y}|+|\\mathbf{Y}-\\mathbf{Z}|. $$"}
{"_id": "289", "text": "If $\\mathbf{X}$ and $\\mathbf{Y}$ are  in $\\R^n,$ then $$ |\\mathbf{X}-\\mathbf{Y}|\\ge\\left| |\\mathbf{X}|-|\\mathbf{Y}|\\right|. $$"}
{"_id": "290", "text": "Under the assumptions of Theorem~$\\ref{thmtype:5.4.3},$ \\begin{equation} \\label{eq:5.4.8} \\frac{\\partial h(\\mathbf{U}_0)}{\\partial u_i}=\\sum_{j=1}^n \\frac{\\partial f(\\mathbf{X}_0) }{\\partial x_j} \\frac{\\partial g_j(\\mathbf{U}_0)}{\\partial u_i},\\quad 1\\le i \\le m. \\end{equation}"}
{"_id": "291", "text": "If $f_{x_1},$ $f_{x_2},$ \\dots$,$ $f_{x_n}$ are identically zero in an open region $S$ of $\\R^n,$ then $f$ is constant in $S.$"}
{"_id": "292", "text": "Suppose that  $f,$ $f_x,$ and $f_y$ are differentiable in a neigborhood of a critical point $\\mathbf{X}_0=(x_0,y_0)$ of $f$ and $f_{xx},$ $f_{yy},$ and $f_{xy}$ are continuous at $(x_0,y_0).$ Let $$ D=f_{xx}(x_0,y_0)f_{xy}(x_0,y_0)-f^2_{xy}(x_0,y_0). $$ Then \\begin{alist} \\item % (a)  $(x_0,y_0)$ is a local extreme point of $f$ if $D>0;$ $(x_0,y_0)$ is a local minimum point if $f_{xx}(x_0,y_0)>0$, or a local maximum point if $f_{xx}(x_0,y_0)<0.$ \\item % (b) $(x_0,y_0)$ is not a local extreme point of $f$ if $D<0.$ \\end{alist}"}
{"_id": "293", "text": "If $\\mathbf{F}$ is continuously differentiable on a neighborhood  of $\\mathbf{X}_0$ and $J\\mathbf{F}(\\mathbf{X}_0)\\ne 0,$ then there is an open neighborhood $N$ of $\\mathbf{X}_0$ on which the conclusions of Theorem~$\\ref{thmtype:6.3.4}$ hold$.$"}
{"_id": "294", "text": "Suppose that  $f:\\R^{n+1}\\to \\R$ is continuously differentiable on an open set containing $(\\mathbf{X}_0,u_0),$ with $f(\\mathbf{X}_0,u_0)=0$ and $f_u(\\mathbf{X}_0,u_0)\\ne0$. Then there is a neighborhood $M$ of $(\\mathbf{X}_0,u_0),$ contained in $S,$ and a neighborhood $N$ of $\\mathbf{X}_0$ in $\\R^n$ on which is defined a unique continuously differentiable function $u=u(\\mathbf{X}):\\R^n\\to \\R$ such that $$ (\\mathbf{X},u(\\mathbf{X}))\\in M\\mbox{\\quad and \\quad}  f_u(\\mathbf{X},u(\\mathbf{X}))\\ne0,\\quad\\mathbf{X}\\in N, $$ $$ u(\\mathbf{X}_0)=u_0, \\mbox{\\quad and \\quad} f(\\mathbf{X},u(\\mathbf{X}))=0,\\quad\\mathbf{X}\\in N. $$ The partial derivatives of $u$ are given by $$ u_{x_i}(\\mathbf{X})=-\\frac{f_{x_i}(\\mathbf{X},u(\\mathbf{X}))}{ f_u(\\mathbf{X},u(\\mathbf{X}))},\\quad 1\\le i\\le n. $$"}
{"_id": "295", "text": "Suppose that  $f$ is integrable on sets $S_1$ and $S_2$ such that $S_1\\cap S_2$ has zero content$.$ Then $f$ is integrable on $S_1\\cup S_2,$ and $$ \\int_{S_1\\cup S_2} f(\\mathbf{X})\\,d\\mathbf{X}= \\int_{S_1} f(\\mathbf{X})\\,d\\mathbf{X}+ \\int_{S_2} f(\\mathbf{X})\\,d\\mathbf{X}. $$"}
{"_id": "296", "text": "If $f$ is integrable on $[a,b] \\times [c,d],$ then $$ \\int_a^b dx\\int_c^d f(x,y)\\,dy=\\int_c^d dy\\int_a^b f(x,y)\\,dx, $$ provided that $\\int_c^d f(x,y)\\,dy$ exists for $a\\le x\\le b$ and $\\int_a^b f(x,y)\\,dx$ exists for $c\\le y\\le d.$ In particular$,$ these hypotheses hold if $f$ is continuous on $[a,b]\\times [c,d].$"}