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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 |
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 ∀s ∈ S, s≤M for some M ∈ R\\ **Completeness Axiom.** If S is a nonempty bounded set in R, then inf S and sup S exist.\\ **Archimedean Property.** If a, b > 0, then ∃n such that na > 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 of S is the greatest lower bound.\\ The sup of S is the smallest upper bound.\\ S is bounded if ∀s ∈ S, s≤M for some M ∈ R\\ **Completeness Axiom.** If S is a nonempty bounded set in R, then inf S and sup S exist.\\ **Archimedean Property.** If a, b > 0, then ∃n such that na > b. |
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A //sequence// (s<sub>n</sub>) is a function mapping from N to R. It converges to s if ∀ ϵ > 0 there exists N such that N > n ⟹ |(s<sub>n</sub>)-s| < ϵ\\ In other words, | A //sequence// (s<sub>n</sub>) is a function mapping from N to R. It converges to s if ∀ $\epsilon$ $>$ 0 there exists N such that N $>$ n ⟹ |(s<sub>n</sub>)-s| $<$ ϵ\\ In other words, |
lim(s<sub>n</sub>) = s | lim(s<sub>n</sub>) $=$ s |
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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(np1) = 0 for p > 0\\ lim n<sup>(1/n)</sup> = 1 | 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(np1) $=$ 0 for p > 0\\ lim n<sup>(1/n)</sup> $=$ 1 |
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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//. |
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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>) = N→∞lim sup{(s<sub>n</sub>): n > N} = sup S\\ | 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>) $=$ N→∞lim sup{(s<sub>n</sub>): n > N} $=$ sup S\\ |
lim inf (s<sub>n</sub>) = N→∞lim inf{(s<sub>n</sub>): n > N} = inf S | lim inf (s<sub>n</sub>) $=$ N→∞lim inf{(s<sub>n</sub>): n > N} $=$ inf S |
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lim inf |s<sub>n+1</sub>| / |s<sub>n</sub>| ≤ lim inf |s<sub>n</sub>|^(1/n) ≤ lim sup |s<sub>n</sub>|^(1/n) ≤ lim sup |s<sub>n+1</sub>| / |s<sub>n</sub>| | lim inf |s<sub>n+1</sub>| / |s<sub>n</sub>| ≤ lim inf |s<sub>n</sub>|^(1/n) ≤ lim sup |s<sub>n</sub>|^(1/n) ≤ lim sup |s<sub>n+1</sub>| / |s<sub>n</sub>| |
**2. Topology** | **2. Topology** |
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**Metric Space:** A set S with a //metric//, distance function d. For any x,y,z ∈ S \\ (1) d(x,y) > 0 if x = y, d(x,x) = 0 \\ (2) d(x,y) = d(y,x)\\ (3) d(x,z) ≤ 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) = ∞ is not valid. | **Metric Space:** A set S with a //metric//, distance function d. For any x,y,z ∈ S \\ (1) d(x,y) $>$ 0 if x = y, d(x,x) $=$ 0 \\ (2) d(x,y) $=$ d(y,x)\\ (3) d(x,z) ≤ 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) $=$ ∞ is not valid. |
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A //limit point// p of a set S is such that for some ϵ radius ball around p, there exists an element q = p such that q ∈/ 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 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 radius ball around p, there exists an element q = p such that q ∈/ 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 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. |
**3. Series** | **3. Series** |
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A series //converges// if its partial sum Σ s<sub>n</sub> = M for some M ∈ R. \\ | A series //converges// if its partial sum Σ s<sub>n</sub> $=$ M for some M ∈ R. \\ |
A series Σ s<sub>n</sub> satisfies is //Cauchy// if for any ϵ > 0, there exists N such that n ≥ m > N ⟹ |i=m∑n s<sub>i</sub>| < ϵ \\ | A series Σ s<sub>n</sub> satisfies is //Cauchy// if for any $\epsilon$ $>$ 0, there exists N such that n ≥ m $>$ N ⟹ |i=m∑n s<sub>i</sub>| $<$ ϵ \\ |
If Σ s<sub>n</sub> converges, then lim s<sub>n</sub> = 0. (Note this is a one-way statement) | If Σ s<sub>n</sub> converges, then lim s<sub>n</sub> $=$ 0. (Note this is a one-way statement) |
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**Comparison Test** \\ | **Comparison Test** \\ |
If Σ a<sub>n</sub> converges and |b<sub>n</sub>| ≤ a<sub>n</sub> ∀n, then Σ b<sub>n</sub> converges. \\ If Σ a<sub>n</sub> = ∞ and |b<sub>n</sub>| ≥ a<sub>n</sub> ∀n, then Σ b<sub>n</sub> = ∞. | If Σ a<sub>n</sub> converges and |b<sub>n</sub>| ≤ a<sub>n</sub> ∀n, then Σ b<sub>n</sub> converges. \\ If Σ a<sub>n</sub> $=$ ∞ and |b<sub>n</sub>| ≥ a<sub>n</sub> ∀n, then Σ b<sub>n</sub> $=$ ∞. |
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**Ratio Test** Σ a<sub>n</sub> of nonzero terms \\ | **Ratio Test** Σ a<sub>n</sub> of nonzero terms \\ |
(i) converges if lim sup |a<sub>n+1</sub> / a<sub>n</sub>| < 1 \\ | (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 \\ | (ii) diverges if lim inf |a<sub>n+1</sub> / a<sub>n</sub>| $>$ 1 \\ |
(iii) else inconclusive test | (iii) else inconclusive test |
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**Root Test** Let α = lim sup |a<sub>n</sub>|<sup>(1/n)</sup>. Then Σ a<sub>n</sub> \\ | **Root Test** Let $\alpha$ $=$ lim sup |a<sub>n</sub>|<sup>(1/n)</sup>. Then Σ a<sub>n</sub> \\ |
(i) converges if α < 1 \\ | (i) converges if $\alpha$ $<$ 1 \\ |
(ii) diverges if α > 1 \\ | (ii) diverges if $\alpha$ $>$ 1 \\ |
(iii) else inconclusive test | (iii) else inconclusive test |
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**Alternating Series Theorem** If a<sub>n</sub> is monotonically decreasing and lim a<sub>n</sub> = 0, then Σ (-1)<sup>n</sup> a<sub>n</sub> converges. | **Alternating Series Theorem** If a<sub>n</sub> is monotonically decreasing and lim a<sub>n</sub> $=$ 0, then Σ (-1)<sup>n</sup> a<sub>n</sub> converges. |
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**4. Continuity and Convergence** | **4. Continuity and Convergence** |
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There are three main definitions for a continuous function f: \\ | There are three main definitions for a continuous function f: \\ |
(1) f continuous at x if for each ϵ > 0, there exists δ > 0 such that |x-y| < δ where y ∈ domain(f) ⟹ |f(x) - f(y)| < ϵ \\ | (1) f continuous at x if for each $\epsilon$ $>$ 0, there exists $\delta$ $>$ 0 such that |x-y| $<$ δ where y ∈ domain(f) ⟹ |f(x) - f(y)| $<$ ϵ \\ |
(2) 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) \\ | (2) 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 f be mapping between metric spaces X → Y. f is continuous if the preimage f<sup>-1</sup> of any open set in Y is open in X. Similarly if the preimage f<sup>-1</sup> of any closed set in Y is closed in X. | (3) Let f be mapping between metric spaces X → Y. f is continuous if the preimage f<sup>-1</sup> of any open set in Y is open in X. Similarly if the preimage f<sup>-1</sup> of any closed set in Y is closed in X. |
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A function f is //uniformly continuous// if for **all x in domain(f)**, for each ϵ > 0, there exists δ > 0 such that |x-y| < δ where y ∈ domain(f) ⟹ |f(x) - f(y)| < ϵ. | A function f is //uniformly continuous// if for **all x in domain(f)**, for each $\epsilon$ $>$ 0, there exists $\delta$ $>$ 0 such that |x-y| $<$ δ where y ∈ domain(f) ⟹ |f(x) - f(y)| $<$ ϵ. |
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Generally, a continuous function f sends a compact set to another compact set. In this case, f is bounded, and sup f and inf f exists. \\ A continuous function acting on a compact set is uniformly continuous on this interval. \\ | Generally, a continuous function f sends a compact set to another compact set. In this case, f is bounded, and sup f and inf f exists. \\ A continuous function acting on a compact set is uniformly continuous on this interval. \\ |
Generally, a continuous function f sends connected set to another connected set. | Generally, a continuous function f sends connected set to another connected set. |
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**Intermediate Value Theorem.** Let 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 ∈ (a,b) such that f(x) = y. | **Intermediate Value Theorem.** Let 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 ∈ (a,b) such that f(x) $=$ y. |
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If a function f is //discontinuous// at x, and both f(x+) and f(x-) exist, then f is said to have discontinuity of the //first kind// at x, or //simple// discontinuity. | If a function f is //discontinuous// at x, and both f(x+) and f(x-) exist, then f is said to have discontinuity of the //first kind// at x, or //simple// discontinuity. |
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A function f<sub>n</sub> converges to f //pointwise// if for each x in the domain, n→∞lim f<sub>n</sub>(x) = f(x). | A function f<sub>n</sub> converges to f //pointwise// if for each x in the domain, n→∞lim f<sub>n</sub>(x) $=$ f(x). |
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A function f<sub>n</sub> converges to f //uniformly// if lim sup{|f<sub>n</sub> - f|} = 0.\\ In other words, for any fixed ϵ > 0 there exists N such that n > N ⟹ |f<sub>n</sub>(x) - f(x)| < ϵ 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<sub>n</sub> converges to f //uniformly// if lim sup{|f<sub>n</sub> - f|} $=$ 0.\\ In other words, for any fixed $\epsilon$ $>$ 0 there exists N such that n $>$ N ⟹ |f<sub>n</sub>(x) - f(x)| $<$ ϵ 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 ⟺ Uniform Convergence |
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| **Weierstrass M-Test.** Let f = ∑i=1∞ f<sub>n</sub>. If ∃M<sub>n</sub> > 0 such that sup |f<sub>n</sub>| ≤ M<sub>n</sub> and ∑i=1∞ M<sub>n</sub> < ∞, then ∑i=1∞ f<sub>n</sub> converges uniformly. |
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| Let K be a compact set, and the following: \\ |
| * {f<sub>n</sub>} is sequence of monotonically decreasing, continuous functions \\ |
| * {f<sub>n</sub>} → f pointwise \\ |
| Then f<sub>n</sub> → f uniformly. |
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| **5. Differentiation and Integration** |
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| Let f be real-valued on [a,b]. Its //derivative// is defined as \\ |
| $\bold{f'(x)}=\lim\limits_{t \to x}\frac{\bold{f(t)} - \bold{f(x)}}{\bold{t - x}}$ |
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| If f is differentiable at x, then f is also continuous at x. \\ |
| If f is differentiable on interval I, and g is differentiable on range(f), then h = g(f) is differentiable on I |
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| A real function f has a //local maximum// at point p if there exists δ > 0 such that f(y) ≤ f(x) for any y where d(x,y) < δ. \\ |
| If f has a local maximum at x, and if $\bold{f'(x)}exists,then\bold{f'(x)}=$ 0. |
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| **Mean Value Theorem.** If f is a real continuous function on [a,b], and is differentiable on (a,b), then there exists an x ∈ (a,b) such that \\ |
| f(b) − f(a) = (b − a) $\bold{f'(x)}Thegeneralizedtheoremfor\bold{f}and\bold{g}$ continuous real functions on [a,b] is \\ |
| (f(b) − f(a)) $\bold{g'(x)}=(\bold{g(b)}-\bold{g(a)})\bold{f'(x)}$ |
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| **Theorem 5.12.** Suppose f is real differentiable function on [a,b], and $\bold{f'(a)}<\lambda<\bold{f'(b)}.Thenthereexistsx\in(a,b)suchthat\bold{f'(x)}=\lambda$. |
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| A function f is said to be //smooth// on interval I if ∀ x ∈ I, ∀ k ∈ N, fk exists. |
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| **L'Hopital Rule.** x→alim g(x)f(x) = x→alim $\frac{\bold{f'(x)}}{\bold{g'(x)}}$ if either |
| * f(x) → 0 and g(x) → 0 as x → a |
| * g(x) → ∞ as x → a |
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| **Taylor's Theorem.** Let f be a real function on [a,b], assume fn−1 is continuous and fn exists, and for any distinct α, β ∈ [a,b] define \\ |
| P(t) = k=0∑n−1 k!fk(α) (t − α)<sup>k</sup> \\ |
| Then there exists a point x between α and β such that \\ |
| f(β) = P(β) + n!fn(x) (β − α)<sup>n</sup> \\ |
| Note Taylor Series on smooth functions may not converge, and may not be equal to original function f(x). |
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| A //partition// P of [a,b] is the finite set of points where a=x<sub>0</sub>≤x<sub>1</sub>≤...x<sub>n</sub>=b \\ |
| Let α be a weight function that is monotonically increasing. Define \\ |
| U(P, f, α) = i=0∑n M<sub>i</sub> Δα<sub>i</sub> \\ |
| L(P, f, α) = i=0∑n m<sub>i</sub> Δα<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, α) = sup L(P, f, α) over all partitions, then the //Riemann integral// of f with respect to α on [a,b] exists \\ |
| ∫abf(x)dα(x) |
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| A //refinement// Q of P contains all the partition points in P, with additional points. \\ |
| If U(P, f, α) − L(P, f, α) < ϵ, then U(Q, f, α) − L(Q, f, α) < ϵ. In other words, refinements maintain the condition for integrability. |
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| **//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 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 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 α(x) = I(x-s) where I is the //unit step function//, then ∫abfdα = f(s) |
| * Suppose α increases monotonically, α' is integrable on [a,b], and f is bounded real function on [a,b]. Then f is integrable with respect to α if and only if fα' is integrable: \\ ∫abfdα = $\int_a^b f(x){\alpha}'(x)d(x)$ |
| * Let f be integrable on [a,b] and for a≤x≤b, let F(x) = ∫axf(t)dt, then \\ (1) F(x) is continuous on [a,b] \\ (2) if f(x) is continuous at p ∈ [a,b], then F(x) is differentiable at p, with F'(p) = f(p) |
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| **Fundamental Theorem of Calculus.** Let f be integrable on [a,b] and F be a differentiable function on [a,b] such that $F'(x)=f(x),then\int_a^b f(x)dx=F(b)-F(a)$ |
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| Let α be increasing weight function on [a,b]. Suppose fn is integrable, and fn → f uniformly on [a,b]. Then f is integrable, and \\ |
| ∫abfdα = n→∞lim ∫abfndα |
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| Suppose {fn} is a sequence of differentiable functions on [a,b] such that fn → g uniformly and there exists p ∈ [a,b] where {fn(p)} converges. Then fn 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. |
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| ==== Questions ==== |
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| **1. What ** |
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