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--- math104-s21:s:vpak [2021/05/07 17:04]
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+++ math104-s21:s:vpak [2022/01/11 10:57] (current)
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-gotta start somewhere
+==== vpak ====
+Hi welcome to my note page
+==== Summary of Material ====
+**1. Numbers, Sets, and Sequences**
+**Rational Zeros Theorem.** For polynomials of the form c<sub>n</sub>x<sup>n</sup> + ... + c<sub>0</sub> = 0 ,
+where each coefficient is an integer, then the //only// rational solutions have the form  $\frac{c}{d}$  where c divides c<sub>n</sub> and d divides c<sub>0</sub>; rational root r must divide c<sub>0</sub>.
+The maximum of a set S is the largest element in the set.\\ The minimum is the smallest element in the set.\\  The  $\inf$  of S is the greatest lower bound.\\  The  $\sup$  of S is the smallest upper bound.\\ S is bounded if  $\forall$ s  $\in$  S, s $\leq$ M for some M  $\in$   $\reals$ \\ **Completeness Axiom.** If S is a nonempty bounded set in  $\reals$ , then  $\inf$  S and  $\sup$  S exist.\\ **Archimedean Property.** If a, b  $\gt$  0, then  $\exists$ n such that na  $\gt$  b.
+A //sequence// (s<sub>n</sub>) is a function mapping from  $\N$  to  $\R$ . It converges to s if  $\forall$   $\epsilon$   $>$  0 there exists N such that N  $>$  n  $\implies$  |(s<sub>n</sub>)-s|  $<$   $\epsilon$ \\ In other words,
+ $\lim$ (s<sub>n</sub>)  $=$  s
+Important limit theorems include:\\  $\lim$ (s<sub>n</sub>)(t<sub>n</sub>)  $=$  ( $\lim$ (s<sub>n</sub>))( $\lim$ (t<sub>n</sub>))\\  $\lim$ (s<sub>n</sub>)+(t<sub>n</sub>)  $=$  ( $\lim$ (s<sub>n</sub>)) + ( $\lim$ (t<sub>n</sub>))\\  $\lim$ ( $\frac{1}{n^p}$ )  $=$  0 for p > 0\\  $\lim$  n<sup>(1/n)</sup>  $=$  1
+A //subsequence// (s<sub>n(k)</sub>) of (s<sub>n</sub>) is a sequence that is a subset of the elements in the original sequence with relative order preserved.\\ **Bolzano-Weierstrass Theorem.** Every bounded sequence has a convergent subsequence, having some //subsequential limit//.
+Given any (s<sub>n</sub>) and let S be the set of subsequential limits of (s<sub>n</sub>). Define:\\  $\lim$   $\sup$  (s<sub>n</sub>)  $=$   $\lim\limits_{N \to \infin}$   $\sup$ {(s<sub>n</sub>): n > N}  $=$   $\sup$  S\\
+ $\lim$   $\inf$  (s<sub>n</sub>)  $=$   $\lim\limits_{N \to \infin}$   $\inf$ {(s<sub>n</sub>): n > N}  $=$   $\inf$  S
+ $\lim$   $\inf$  |s<sub>n+1</sub>| / |s<sub>n</sub>|  $\leq$   $\lim$   $\inf$  |s<sub>n</sub>|^(1/n)  $\leq$   $\lim$   $\sup$  |s<sub>n</sub>|^(1/n)  $\leq$   $\lim$   $\sup$  |s<sub>n+1</sub>| / |s<sub>n</sub>|
+**2. Topology**
+**Metric Space:** A set S with a //metric//, distance function d. For any x,y,z  $\in$  S \\ (1) d(x,y)  $>$  0 if x  $\not =$  y, d(x,x)  $=$  0 \\ (2) d(x,y)  $=$  d(y,x)\\ (3) d(x,z)  $\leq$  d(x,y) + d(y,z) \\ **Important* **A metric is only valid if it outputs a real number for any inputs, ie. d(x,y)  $=$   $\infin$  is not valid.
+A //limit point// p of a set S is such that for some  $\epsilon$  radius ball around p, there exists an element q  $\not =$  p such that q  $\notin$  S. Note that limit points may or may not lie in the set. \\ A set S is //open// if for every point p in S is interior in S. Think open ball of  $\epsilon$  radius in S centered at p. \\ A set S is //closed// if every limit point of S is a point in S. \\ A set S is //perfect// if it is closed and every interior point is a limit point. \\ A set S is //dense// in a metric space X if every point in X is either a limit point of S or in S itself. \\ The //closure// of a set S is the union of S and the set of its limit points. It can also be thought of as the intersection of all closed sets containing S.
+An //open cover// for S is a collection of open sets that covers S. \\
+A set S is //compact// if for all open covers {G} of S, there exists a finite subcover of {G} that covers S. \\ **Heine-Borel Theorem.** A subset E of  $\R$ <sup>n</sup> is compact //if and only if// it is closed and bounded.
+A set S is //connected// if it **cannot** be written as disjoint union of nonempty, open sets
+**3. Series**
+A series //converges// if its partial sum  $\Sigma$  s<sub>n</sub>  $=$  M for some M  $\in$   $\reals$ . \\
+A series  $\Sigma$  s<sub>n</sub> satisfies is //Cauchy// if for any  $\epsilon$   $>$  0, there exists N such that n  $\geq$  m  $>$  N  $\implies$  | $\displaystyle\sum_{i=m}^n$  s<sub>i</sub>|  $<$   $\epsilon$   \\
+If  $\Sigma$  s<sub>n</sub> converges, then  $\lim$  s<sub>n</sub>  $=$  0. (Note this is a one-way statement)
+**Comparison Test** \\
+If  $\Sigma$  a<sub>n</sub> converges and |b<sub>n</sub>|  $\leq$  a<sub>n</sub>  $\forall$ n, then  $\Sigma$  b<sub>n</sub> converges. \\ If  $\Sigma$  a<sub>n</sub>  $=$   $\infin$  and |b<sub>n</sub>|  $\geq$  a<sub>n</sub>  $\forall$ n, then  $\Sigma$  b<sub>n</sub>  $=$   $\infin$ .
+**Ratio Test**   $\Sigma$  a<sub>n</sub> of nonzero terms \\
+(i) converges if  $\lim$   $\sup$  |a<sub>n+1</sub> / a<sub>n</sub>|  $<$  1 \\
+(ii) diverges if  $\lim$   $\inf$  |a<sub>n+1</sub> / a<sub>n</sub>|  $>$  1 \\
+(iii) else inconclusive test
+**Root Test** Let  $\alpha$   $=$   $\lim$   $\sup$  |a<sub>n</sub>|<sup>(1/n)</sup>. Then  $\Sigma$  a<sub>n</sub> \\
+(i) converges if  $\alpha$   $<$  1 \\
+(ii) diverges if  $\alpha$   $>$  1 \\
+(iii) else inconclusive test
+**Alternating Series Theorem**  If a<sub>n</sub> is monotonically decreasing and  $\lim$  a<sub>n</sub>  $=$  0, then  $\Sigma$  (-1)<sup>n</sup> a<sub>n</sub> converges.
+**4. Continuity and Convergence**
+There are three main definitions for a continuous function  $\bold{f}$ : \\
+(1)  $\bold{f}$  continuous at x if for each  $\epsilon$   $>$  0, there exists  $\delta$   $>$  0 such that |x-y|  $<$   $\delta$  where y  $\in$  domain(f)  $\implies$  |f(x) - f(y)|  $<$   $\epsilon$  \\
+(2)  $\bold{f}$  continuous at x if for all sequences (s<sub>n</sub>) in domain(f) that converge to x,  $\lim$  f(s<sub>n</sub>)  $=$  f(x) \\
+(3) Let  $\bold{f}$  be mapping between metric spaces X  $\to$  Y.  $\bold{f}$  is continuous if the preimage  $\bold{f}$ <sup>-1</sup> of any open set in Y is open in X. Similarly if the preimage  $\bold{f}$ <sup>-1</sup> of any closed set in Y is closed in X.
+A  function  $\bold{f}$  is //uniformly continuous// if for **all x in domain(f)**, for each  $\epsilon$   $>$  0, there exists  $\delta$   $>$  0 such that |x-y|  $<$   $\delta$  where y  $\in$  domain(f)  $\implies$  |f(x) - f(y)|  $<$   $\epsilon$ .
+Generally, a continuous function  $\bold{f}$  sends a compact set to another compact set. In this case,  $\bold{f}$  is bounded, and  $\sup$   $\bold{f}$  and  $\inf$   $\bold{f}$  exists. \\ A continuous function acting on a compact set is uniformly continuous on this interval. \\
+//Cauchy relation:// If  $\bold{f}$  is uniformly continuous on a set S and s<sub>n</sub> is a Cauchy sequence in S, then  $\bold{f}$ (s<sub>n</sub>) is a Cauchy sequence.
+Generally, a continuous function  $\bold{f}$  sends connected set to another connected set.
+**Intermediate Value Theorem.** Let  $\bold{f}$  be a continuous real function on interval [a,b]. Then for all y between f(a) and f(b), there exists at least one x  $\in$  (a,b) such that f(x)  $=$  y.
+If a function  $\bold{f}$  is //discontinuous// at x, and both  $\bold{f}$ (x+) and  $\bold{f}$ (x-) exist, then  $\bold{f}$  is said to have discontinuity of the //first kind// at x, or //simple// discontinuity.
+A function  $\bold{f}$ <sub>n</sub> converges to f //pointwise// if for each x in the domain,  $\lim\limits_{n \to \infin}$   $\bold{f}$ <sub>n</sub>(x)  $=$  f(x).
+A function  $\bold{f}$ <sub>n</sub> converges to f //uniformly// if  $\lim$   $\sup$ {| $\bold{f}$ <sub>n</sub> - f|}  $=$  0.\\  In other words, for any fixed  $\epsilon$   $>$  0 there exists N such that n  $>$  N  $\implies$  | $\bold{f}$ <sub>n</sub>(x) - f(x)|  $<$   $\epsilon$  for all x in the domain. This is a stronger restraint than pointwise convergence, where we only cared about each x individually. (Compare with regular vs uniform continuity) \\ Uniform Cauchy  $\iff$  Uniform Convergence
+**Weierstrass M-Test.** Let  $\bold{f}$   $\ =$   $\textstyle\sum_{i=1}^\infin$   $\bold{f}$ <sub>n</sub>. If  $\exists$ M<sub>n</sub>  $>$  0 such that  $\sup$  | $\bold{f}$ <sub>n</sub>|  $\leq$  M<sub>n</sub> and  $\textstyle\sum_{i=1}^\infin$  M<sub>n</sub>  $<$   $\infin$ , then  $\textstyle\sum_{i=1}^\infin$   $\bold{f}$ <sub>n</sub> converges uniformly.
+Let K be a compact set, and the following: \\
+  * { $\bold{f}$ <sub>n</sub>} is sequence of monotonically decreasing, continuous functions \\
+  * { $\bold{f}$ <sub>n</sub>}  $\to$   $\bold{f}$  pointwise \\
+Then  $\bold{f}$ <sub>n</sub>  $\to$   $\bold{f}$  uniformly.
+**5. Differentiation and Integration**
+Let  $\bold{f}$  be real-valued on [a,b]. Its //derivative// is defined as \\
+$\bold{f'(x)}=\lim\limits_{t \to x}\frac{\bold{f(t)} - \bold{f(x)}}{\bold{t - x}}$
+If  $\bold{f}$  is differentiable at x, then  $\bold{f}$  is also continuous at x. \\
+If  $\bold{f}$  is differentiable on interval  $\bold{I}$ , and  $\bold{g}$  is differentiable on range( $\bold{f}$ ),  then  $\bold{h}$   $=$   $\bold{g(\bold{f})}$  is differentiable on  $\bold{I}$
+A real function  $\bold{f}$  has a //local maximum// at point p if there exists  $\delta$   $>$  0 such that  $\bold{f(y)}$   $\leq$   $\bold{f(x)}$  for any y where d(x,y)  $<$   $\delta$ . \\
+If  $\bold{f}$  has a local maximum at x, and if $\bold{f'(x)} $exists, then$ \bold{f'(x)}=$ 0.
+**Mean Value Theorem.** If  $\bold{f}$  is a real continuous function on [a,b], and is differentiable on (a,b), then there exists an x  $\in$  (a,b) such that \\
+ $\bold{f(b)}$   $-$   $\bold{f(a)}$   $=$  (b  $-$  a) $\bold{f'(x)} $\\ The generalized theorem for$ \bold{f} $and$ \bold{g}$ continuous real functions on [a,b] is \\
+( $\bold{f(b)}$   $-$   $\bold{f(a)}$ ) $\bold{g'(x)}= $($ \bold{g(b)}-\bold{g(a)} $)$ \bold{f'(x)}$
+**Theorem 5.12.** Suppose  $\bold{f}$  is real differentiable function on [a,b], and $\bold{f'(a)}<\lambda<\bold{f'(b)} $. Then there exists x$ \in $(a,b) such that$ \bold{f'(x)}=\lambda$.
+A function  $\bold{f}$  is said to be //smooth// on interval I if  $\forall$  x  $\in$  I,  $\forall$  k  $\in$   $\N$ ,  $\bold{f^k}$  exists.
+**L'Hopital Rule.**  $\lim\limits_{x \to a}$   $\frac{\bold{f(x)}}{\bold{g(x)}}$   $=$   $\lim\limits_{x \to a}$  $\frac{\bold{f'(x)}}{\bold{g'(x)}}$ if either
+  *  $\bold{f(x)}$   $\to$  0 and  $\bold{g(x)}$   $\to$  0 as x  $\to$  a
+  *  $\bold{g(x)}$   $\to$   $\infin$  as x  $\to$  a
+**Taylor's Theorem.** Let  $\bold{f}$  be a real function on [a,b], assume  $\bold{f^{n-1}}$  is continuous and  $\bold{f^n}$  exists, and for any distinct  $\alpha$ ,  $\beta$   $\in$  [a,b] define \\
+P(t)  $=$   $\displaystyle\sum_{k=0}^{n-1}$   $\frac{\bold{f^k(\alpha)}}{k!}$  (t  $-$   $\alpha$ )<sup>k</sup> \\
+Then there exists a point x between  $\alpha$  and  $\beta$  such that \\
+ $\bold{f(\beta)}$   $=$  P( $\beta$ )  $+$   $\frac{\bold{f^n(x)}}{n!}$  ( $\beta$   $-$   $\alpha$ )<sup>n</sup> \\
+Note Taylor Series on smooth functions may not converge, and may not be equal to original function f(x).
+A //partition// P of [a,b] is the finite set of points where a $=$ x<sub>0</sub> $\leq$ x<sub>1</sub> $\leq$ ...x<sub>n</sub> $=$ b \\
+Let  $\alpha$  be a weight function that is monotonically increasing. Define \\
+U(P, f,  $\alpha$ )  $=$   $\displaystyle\sum_{i=0}^{n}$  M<sub>i</sub>  $\Delta{\alpha}$ <sub>i</sub> \\
+L(P, f,  $\alpha$ )  $=$   $\displaystyle\sum_{i=0}^{n}$  m<sub>i</sub>  $\Delta{\alpha}$ <sub>i</sub> \\
+where M<sub>i</sub> is the  $\sup$  and m<sub>i</sub> is the  $\inf$  over that subinterval. \\
+If  $\inf$  U(P, f,  $\alpha$ )  $=$   $\sup$  L(P, f,  $\alpha$ ) over all partitions, then the //Riemann integral// of f with respect to  $\alpha$  on [a,b] exists \\
+ $\int_a^b f(x)d{\alpha}(x)$
+A //refinement// Q of P contains all the partition points in P, with additional points. \\
+If U(P, f,  $\alpha$ )  $-$  L(P, f,  $\alpha$ )  $<$   $\epsilon$ , then U(Q, f,  $\alpha$ )  $-$  L(Q, f,  $\alpha$ )  $<$   $\epsilon$ . In other words, refinements maintain the condition for integrability.
+**//Key Theorems: //** \\
+  * If f is continuous on [a,b], then f is integrable on [a,b].
+  * If f is monotonic on [a,b] and if  $\alpha$  is continuous on [a,b], then f is integrable on [a,b].
+  * Suppose f is bounded and has finitely many discontinuities on [a,b]. If  $\alpha$  is continuous at every point of discontinuity, then f is integrable.
+  * If f is integrable on [a,b] and g is continuous on the range of f, then h  $=$  g(f) is integrable on [a,b].
+  * If a$< $s$ <$b, f is bounded, f is continuous at s, and  $\alpha$ (x)  $=$  I(x-s) where I is the //unit step function//, then  $\int_a^b fd{\alpha}$   $=$  f(s)
+  * Suppose  $\alpha$  increases monotonically,  $\alpha$ ' is integrable on [a,b], and f is bounded real function on [a,b]. Then f is integrable with respect to  $\alpha$  if and only if f $\alpha$ ' is integrable: \\  $\int_a^b fd{\alpha}$   $=$  $\int_a^b f(x){\alpha}'(x)d(x)$
+  * Let f be integrable on [a,b] and for a $\leq$ x $\leq$ b, let F(x)  $=$   $\int_a^x f(t)dt$ , then \\ (1) F(x) is continuous on [a,b] \\ (2) if f(x) is continuous at p  $\in$  [a,b], then F(x) is differentiable at p, with F'(p)  $=$  f(p)
+**Fundamental Theorem of Calculus.** Let  $f$  be integrable on  $[a,b]$  and  $F$  be a differentiable function on [a,b] such that $F'(x)=f(x) $, then$ \int_a^b f(x)dx=F(b)-F(a)$
+Let  $\alpha$  be increasing weight function on  $[a,b]$ . Suppose  $f_n$  is integrable, and  $f_n$   $\to$   $f$  uniformly on  $[a,b]$ . Then  $f$  is integrable, and \\
+ $\int_a^b fd{\alpha}$   $=$   $\lim\limits_{n \to \infin}$   $\int_a^b f_{n}d{\alpha}$
+Suppose { $f_n$ } is a sequence of differentiable functions on  $[a,b]$  such that  $f_n$   $\to$   $g$  uniformly and there exists p  $\in$   $[a,b]$  where { $f_n(p)$ } converges. Then  $f_n$  converges to some  $f$  uniformly, and \\
+$f'(x)=g(x)=\lim\limits_{n \to \infin}f'_{n}(x)$ \\
+Note $f'_{n}(x)$ may not be continuous.
+==== Questions ====
+**1. What **
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