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math104-s21:s:vpak [2021/05/09 23:18]
68.186.63.173 [Summary of Material]
math104-s21:s:vpak [2022/01/11 10:57] (current)
pzhou ↷ Page moved from math104-2021sp:s:vpak to math104-s21:s:vpak
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 The maximum of a set S is the largest element in the set.\\ The minimum is the smallest element in the set.\\  The inf\inf of S is the greatest lower bound.\\  The sup\sup of S is the smallest upper bound.\\ S is bounded if \foralls \in S, s\leqM for some M \in R\reals\\ **Completeness Axiom.** If S is a nonempty bounded set in R\reals, then inf\inf S and sup\sup S exist.\\ **Archimedean Property.** If a, b >\gt 0, then \existsn such that na >\gt b. The maximum of a set S is the largest element in the set.\\ The minimum is the smallest element in the set.\\  The inf\inf of S is the greatest lower bound.\\  The sup\sup of S is the smallest upper bound.\\ S is bounded if \foralls \in S, s\leqM for some M \in R\reals\\ **Completeness Axiom.** If S is a nonempty bounded set in R\reals, then inf\inf S and sup\sup S exist.\\ **Archimedean Property.** If a, b >\gt 0, then \existsn such that na >\gt b.
  
-A //sequence// (s<sub>n</sub>) is a function mapping from N\N to R\R. It converges to s if \forall ϵ\epsilon > 0 there exists N such that N > n     \implies |(s<sub>n</sub>)-s| < ϵ\epsilon\\ In other words,  +A //sequence// (s<sub>n</sub>) is a function mapping from N\N to R\R. It converges to s if \forall $\epsilon$>0 there exists N such that N $>n     \implies |(s<sub>n</sub>)-s| $<ϵ\epsilon\\ In other words,  
-lim\lim(s<sub>n</sub>) = s+lim\lim(s<sub>n</sub>$=s
  
-Important limit theorems include:\\ lim\lim(s<sub>n</sub>)(t<sub>n</sub>) = (lim\lim(s<sub>n</sub>))(lim\lim(t<sub>n</sub>))\\ lim\lim(s<sub>n</sub>)+(t<sub>n</sub>) = (lim\lim(s<sub>n</sub>)) + (lim\lim(t<sub>n</sub>))\\ lim\lim(1np\frac{1}{n^p}) = 0 for p > 0\\ lim\lim n<sup>(1/n)</sup> = 1+Important limit theorems include:\\ lim\lim(s<sub>n</sub>)(t<sub>n</sub>$=(lim\lim(s<sub>n</sub>))(lim\lim(t<sub>n</sub>))\\ lim\lim(s<sub>n</sub>)+(t<sub>n</sub>$=(lim\lim(s<sub>n</sub>)) + (lim\lim(t<sub>n</sub>))\\ lim\lim(1np\frac{1}{n^p}$=0 for p > 0\\ lim\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// 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\lim sup\sup (s<sub>n</sub>) = limN\lim\limits_{N \to \infin} sup\sup{(s<sub>n</sub>): n > N} = sup\sup S\\  +Given any (s<sub>n</sub>) and let S be the set of subsequential limits of (s<sub>n</sub>). Define:\\ lim\lim sup\sup (s<sub>n</sub>$=limN\lim\limits_{N \to \infin} sup\sup{(s<sub>n</sub>): n > N} $=sup\sup S\\  
-lim\lim inf\inf (s<sub>n</sub>) = limN\lim\limits_{N \to \infin} inf\inf{(s<sub>n</sub>): n > N} = inf\inf S+lim\lim inf\inf (s<sub>n</sub>$=limN\lim\limits_{N \to \infin} inf\inf{(s<sub>n</sub>): n > N} $=inf\inf S
  
 lim\lim inf\inf |s<sub>n+1</sub>| / |s<sub>n</sub>| \leq lim\lim inf\inf |s<sub>n</sub>|^(1/n) \leq lim\lim sup\sup |s<sub>n</sub>|^(1/n) \leq lim\lim sup\sup |s<sub>n+1</sub>| / |s<sub>n</sub>| lim\lim inf\inf |s<sub>n+1</sub>| / |s<sub>n</sub>| \leq lim\lim inf\inf |s<sub>n</sub>|^(1/n) \leq lim\lim sup\sup |s<sub>n</sub>|^(1/n) \leq lim\lim sup\sup |s<sub>n+1</sub>| / |s<sub>n</sub>|
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 **2. Topology** **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.+**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 ϵ\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 ϵ\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. A //limit point// p of a set S is such that for some ϵ\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 ϵ\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.
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 **3. Series** **3. Series**
  
-A series //converges// if its partial sum Σ\Sigma s<sub>n</sub> = M for some M \in R\reals. \\  +A series //converges// if its partial sum Σ\Sigma s<sub>n</sub> $=M for some M \in R\reals. \\  
-A series Σ\Sigma s<sub>n</sub> satisfies is //Cauchy// if for any ϵ\epsilon > 0, there exists N such that n \geq m > N     \implies |i=mn\displaystyle\sum_{i=m}^n s<sub>i</sub>| < ϵ\epsilon  \\  +A series Σ\Sigma s<sub>n</sub> satisfies is //Cauchy// if for any $\epsilon$>0, there exists N such that n \geq$>N     \implies |i=mn\displaystyle\sum_{i=m}^n s<sub>i</sub>$<ϵ\epsilon  \\  
-If Σ\Sigma s<sub>n</sub> converges, then lim\lim s<sub>n</sub> = 0. (Note this is a one-way statement)+If Σ\Sigma s<sub>n</sub> converges, then lim\lim s<sub>n</sub> $=0. (Note this is a one-way statement)
  
 **Comparison Test** \\  **Comparison Test** \\ 
-If Σ\Sigma a<sub>n</sub> converges and |b<sub>n</sub>| \leq a<sub>n</sub> \foralln, then Σ\Sigma b<sub>n</sub> converges. \\ If Σ\Sigma a<sub>n</sub> = \infin and |b<sub>n</sub>| \geq a<sub>n</sub> \foralln, then Σ\Sigma b<sub>n</sub> = \infin+If Σ\Sigma a<sub>n</sub> converges and |b<sub>n</sub>| \leq a<sub>n</sub> \foralln, then Σ\Sigma b<sub>n</sub> converges. \\ If Σ\Sigma a<sub>n</sub> $=\infin and |b<sub>n</sub>| \geq a<sub>n</sub> \foralln, then Σ\Sigma b<sub>n</sub> $=\infin
  
 **Ratio Test**  Σ\Sigma a<sub>n</sub> of nonzero terms \\  **Ratio Test**  Σ\Sigma a<sub>n</sub> of nonzero terms \\ 
-(i) converges if lim\lim sup\sup |a<sub>n+1</sub> / a<sub>n</sub>| < 1 \\  +(i) converges if lim\lim sup\sup |a<sub>n+1</sub> / a<sub>n</sub>$<1 \\  
-(ii) diverges if lim\lim inf\inf |a<sub>n+1</sub> / a<sub>n</sub>| > 1 \\ +(ii) diverges if lim\lim inf\inf |a<sub>n+1</sub> / a<sub>n</sub>$>1 \\ 
 (iii) else inconclusive test (iii) else inconclusive test
  
-**Root Test** Let α\alpha = lim\lim sup\sup |a<sub>n</sub>|<sup>(1/n)</sup>. Then Σ\Sigma a<sub>n</sub> \\  +**Root Test** Let $\alpha$=lim\lim sup\sup |a<sub>n</sub>|<sup>(1/n)</sup>. Then Σ\Sigma a<sub>n</sub> \\  
-(i) converges if α\alpha < 1 \\  +(i) converges if $\alpha$<1 \\  
-(ii) diverges if α\alpha > 1 \\+(ii) diverges if $\alpha$>1 \\
 (iii) else inconclusive test (iii) else inconclusive test
  
-**Alternating Series Theorem**  If a<sub>n</sub> is monotonically decreasing and lim\lim a<sub>n</sub> = 0, then Σ\Sigma (-1)<sup>n</sup> a<sub>n</sub> converges.+**Alternating Series Theorem**  If a<sub>n</sub> is monotonically decreasing and lim\lim a<sub>n</sub> $=0, then Σ\Sigma (-1)<sup>n</sup> a<sub>n</sub> converges.
  
 **4. Continuity and Convergence** **4. Continuity and Convergence**
  
 There are three main definitions for a continuous function f\bold{f}: \\  There are three main definitions for a continuous function f\bold{f}: \\ 
-(1) f\bold{f} continuous at x if for each ϵ\epsilon > 0, there exists δ\delta > 0 such that |x-y| < δ\delta where y \in domain(f)     \implies |f(x) - f(y)| < ϵ\epsilon \\  +(1) f\bold{f} continuous at x if for each $\epsilon$>0, there exists $\delta$>0 such that |x-y| $<δ\delta where y \in domain(f)     \implies |f(x) - f(y)| $<ϵ\epsilon \\  
-(2) f\bold{f} continuous at x if for all sequences (s<sub>n</sub>) in domain(f) that converge to x, lim\lim f(s<sub>n</sub>) = f(x) \\ +(2) f\bold{f} continuous at x if for all sequences (s<sub>n</sub>) in domain(f) that converge to x, lim\lim f(s<sub>n</sub>$=f(x) \\ 
 (3) Let f\bold{f} be mapping between metric spaces X \to Y. f\bold{f} is continuous if the preimage f\bold{f}<sup>-1</sup> of any open set in Y is open in X. Similarly if the preimage f\bold{f}<sup>-1</sup> of any closed set in Y is closed in X.   (3) Let f\bold{f} be mapping between metric spaces X \to Y. f\bold{f} is continuous if the preimage f\bold{f}<sup>-1</sup> of any open set in Y is open in X. Similarly if the preimage f\bold{f}<sup>-1</sup> of any closed set in Y is closed in X.  
  
-A  function f\bold{f} is //uniformly continuous// if for **all x in domain(f)**, for each ϵ\epsilon > 0, there exists δ\delta > 0 such that |x-y| < δ\delta where y \in domain(f)     \implies |f(x) - f(y)| < ϵ\epsilon.+A  function f\bold{f} is //uniformly continuous// if for **all x in domain(f)**, for each $\epsilon$>0, there exists $\delta$>0 such that |x-y| $<δ\delta where y \in domain(f)     \implies |f(x) - f(y)| $<ϵ\epsilon.
  
 Generally, a continuous function f\bold{f} sends a compact set to another compact set. In this case, f\bold{f} is bounded, and sup\sup f\bold{f} and inf\inf f\bold{f} exists. \\ A continuous function acting on a compact set is uniformly continuous on this interval. \\  Generally, a continuous function f\bold{f} sends a compact set to another compact set. In this case, f\bold{f} is bounded, and sup\sup f\bold{f} and inf\inf f\bold{f} exists. \\ A continuous function acting on a compact set is uniformly continuous on this interval. \\ 
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 Generally, a continuous function f\bold{f} sends connected set to another connected set.  Generally, a continuous function f\bold{f} sends connected set to another connected set. 
  
-**Intermediate Value Theorem.** Let f\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.+**Intermediate Value Theorem.** Let f\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 f\bold{f} is //discontinuous// at x, and both f\bold{f}(x+) and f\bold{f}(x-) exist, then f\bold{f} is said to have discontinuity of the //first kind// at x, or //simple// discontinuity.  If a function f\bold{f} is //discontinuous// at x, and both f\bold{f}(x+) and f\bold{f}(x-) exist, then f\bold{f} is said to have discontinuity of the //first kind// at x, or //simple// discontinuity. 
  
-A function f\bold{f}<sub>n</sub> converges to f //pointwise// if for each x in the domain, limn\lim\limits_{n \to \infin} f\bold{f}<sub>n</sub>(x) = f(x). +A function f\bold{f}<sub>n</sub> converges to f //pointwise// if for each x in the domain, limn\lim\limits_{n \to \infin} f\bold{f}<sub>n</sub>(x) $=f(x). 
  
-A function f\bold{f}<sub>n</sub> converges to f //uniformly// if lim\lim sup\sup{|f\bold{f}<sub>n</sub> - f|} = 0.\\  In other words, for any fixed ϵ\epsilon > 0 there exists N such that n > N     \implies |f\bold{f}<sub>n</sub>(x) - f(x)| < ϵ\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)+A function f\bold{f}<sub>n</sub> converges to f //uniformly// if lim\lim sup\sup{|f\bold{f}<sub>n</sub> - f|} $=0.\\  In other words, for any fixed $\epsilon$>0 there exists N such that n $>N     \implies |f\bold{f}<sub>n</sub>(x) - f(x)| $<ϵ\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 f\bold{f}  =\ = i=1\textstyle\sum_{i=1}^\infin f\bold{f}<sub>n</sub>. If \existsM<sub>n</sub> >> 0 such that sup\sup |f\bold{f}<sub>n</sub>| \leq M<sub>n</sub> and i=1\textstyle\sum_{i=1}^\infin M<sub>n</sub> << \infin, then i=1\textstyle\sum_{i=1}^\infin f\bold{f}<sub>n</sub> converges uniformly.      
  
 +Let K be a compact set, and the following: \\ 
 +  * {f\bold{f}<sub>n</sub>} is sequence of monotonically decreasing, continuous functions \\ 
 +  * {f\bold{f}<sub>n</sub>} \to f\bold{f} pointwise \\ 
 +Then f\bold{f}<sub>n</sub> \to f\bold{f} uniformly.
  
 +**5. Differentiation and Integration**
  
 +Let f\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 f\bold{f} is differentiable at x, then f\bold{f} is also continuous at x. \\  
 +If f\bold{f} is differentiable on interval I\bold{I}, and g\bold{g} is differentiable on range(f\bold{f}),  then h\bold{h} == g(f)\bold{g(\bold{f})} is differentiable on I\bold{I} 
 + 
 +A real function f\bold{f} has a //local maximum// at point p if there exists δ\delta >> 0 such that f(y)\bold{f(y)} \leq f(x)\bold{f(x)} for any y where d(x,y) << δ\delta. \\  
 +If f\bold{f} has a local maximum at x, and if $\bold{f'(x)}exists,then exists, then \bold{f'(x)} =$ 0. 
 + 
 +**Mean Value Theorem.** If f\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 \\  
 +f(b)\bold{f(b)} - f(a)\bold{f(a)} == (b - a) $\bold{f'(x)}Thegeneralizedtheoremfor \\ The generalized theorem for \bold{f}and and \bold{g}$ continuous real functions on [a,b] is \\  
 +(f(b)\bold{f(b)} - f(a)\bold{f(a)}) $\bold{g'(x)} =( (\bold{g(b)} - \bold{g(a)})) \bold{f'(x)}$ 
 + 
 +**Theorem 5.12.** Suppose f\bold{f} is real differentiable function on [a,b], and $\bold{f'(a)} < \lambda < \bold{f'(b)}.Thenthereexistsx. Then there exists x \in(a,b)suchthat (a,b) such that \bold{f'(x)} = \lambda$. 
 + 
 +A function f\bold{f} is said to be //smooth// on interval I if \forall x \in I, \forall k \in N\N, fk\bold{f^k} exists.  
 + 
 +**L'Hopital Rule.** limxa\lim\limits_{x \to a} f(x)g(x)\frac{\bold{f(x)}}{\bold{g(x)}} == limxa\lim\limits_{x \to a} $\frac{\bold{f'(x)}}{\bold{g'(x)}}$ if either  
 +  * f(x)\bold{f(x)} \to 0 and g(x)\bold{g(x)} \to 0 as x \to
 +  * g(x)\bold{g(x)} \to \infin as x \to
 + 
 +**Taylor's Theorem.** Let f\bold{f} be a real function on [a,b], assume fn1\bold{f^{n-1}} is continuous and fn\bold{f^n} exists, and for any distinct α\alpha, β\beta \in [a,b] define \\  
 +P(t) == k=0n1\displaystyle\sum_{k=0}^{n-1} fk(α)k!\frac{\bold{f^k(\alpha)}}{k!} (t - α\alpha)<sup>k</sup> \\  
 +Then there exists a point x between α\alpha and β\beta such that \\  
 +f(β)\bold{f(\beta)} == P(β\beta) ++ fn(x)n!\frac{\bold{f^n(x)}}{n!} (β\beta - α\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>\leqx<sub>1</sub>\leq...x<sub>n</sub>==b \\  
 +Let α\alpha be a weight function that is monotonically increasing. Define \\  
 +U(P, f, α\alpha) == i=0n\displaystyle\sum_{i=0}^{n} M<sub>i</sub> Δα\Delta{\alpha}<sub>i</sub> \\  
 +L(P, f, α\alpha) == i=0n\displaystyle\sum_{i=0}^{n} m<sub>i</sub> Δα\Delta{\alpha}<sub>i</sub> \\  
 +where M<sub>i</sub> is the sup\sup and m<sub>i</sub> is the inf\inf over that subinterval. \\  
 +If inf\inf U(P, f, α\alpha) == sup\sup L(P, f, α\alpha) over all partitions, then the //Riemann integral// of f with respect to α\alpha on [a,b] exists \\  
 +abf(x)dα(x)\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, α\alpha) - L(P, f, α\alpha) << ϵ\epsilon, then U(Q, f, α\alpha) - L(Q, f, α\alpha) << ϵ\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 α\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 α\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$<ss<$b, f is bounded, f is continuous at s, and α\alpha(x) == I(x-s) where I is the //unit step function//, then abfdα\int_a^b fd{\alpha} == f(s) 
 +  * Suppose α\alpha increases monotonically, α\alpha' is integrable on [a,b], and f is bounded real function on [a,b]. Then f is integrable with respect to α\alpha if and only if fα\alpha' is integrable: \\ abfdα\int_a^b fd{\alpha} == $\int_a^b f(x){\alpha}'(x)d(x)$ 
 +  * Let f be integrable on [a,b] and for a\leqx\leqb, let F(x) == axf(t)dt\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 ff be integrable on [a,b][a,b] and FF be a differentiable function on [a,b] such that $F'(x) = f(x),then, then \int_a^b f(x)dx = F(b) - F(a)$  
 + 
 +Let α\alpha be increasing weight function on [a,b][a,b]. Suppose fnf_n is integrable, and fnf_n \to ff uniformly on [a,b][a,b]. Then ff is integrable, and \\  
 +abfdα\int_a^b fd{\alpha} == limn\lim\limits_{n \to \infin} abfndα\int_a^b f_{n}d{\alpha} 
 + 
 +Suppose {fnf_n} is a sequence of differentiable functions on [a,b][a,b] such that fnf_n \to gg uniformly and there exists p \in [a,b][a,b] where {fn(p)f_n(p)} converges. Then fnf_n converges to some ff 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 **
  
  
math104-s21/s/vpak.1620627505.txt.gz · Last modified: 2021/05/09 23:18 by 68.186.63.173