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26 April 2024
- 03:0003:00, 26 April 2024 diff hist +1,573 N Limit to Infinity of Binomial Coefficient over Power/Proof 2 Created page with "== Theorem == {{:Limit to Infinity of Binomial Coefficient over Power}} == Proof == <onlyinclude> This proof applies to the special case where $k \in \Z$. Then, {{hypothesis}}, we need only consider: :$k \in \set {0, 1, 2, \dotsc}$ By Gamma Function Extends Factorial, it suffices to show: :$\ds \lim_{r \mathop \to \infty} \frac {\dbinom r k} {r^k} = \frac 1 {k !}$ We have: {{begin-eqn}} {{eqn | q = \forall r \in \R | l = \frac {\dbinom r k} {r^k} |..." current
- 02:4102:41, 26 April 2024 diff hist 0 m Definition:Power (Algebra)/Integer No edit summary current
- 02:3402:34, 26 April 2024 diff hist +59 Limit to Infinity of Binomial Coefficient over Power No edit summary current
- 02:1602:16, 26 April 2024 diff hist 0 m Definition:Binomial Coefficient/Real Numbers No edit summary current
- 02:1402:14, 26 April 2024 diff hist +1,855 N Limit to Infinity of Binomial Coefficient over Power/Proof 1 Created page with "== Theorem == {{:Limit to Infinity of Binomial Coefficient over Power}} == Proof == <onlyinclude> {{begin-eqn}} {{eqn | l = \lim_{r \mathop \to \infty} \frac {\dbinom r k} {r^k} | r = \lim_{r \mathop \to \infty} \frac {\map \Gamma {r + 1} } {\map \Gamma {k + 1} \map \Gamma {r - k + 1} r^k} | c = Gamma Function Extends Factorial }} {{eqn | r = \lim_{r \mathop \to \infty} \frac 1 {\map \Gamma {k + 1} } \frac {\sqrt {2 \pi r} \paren {r / e}^r} {\sqrt {2 \p..." current
- 02:1402:14, 26 April 2024 diff hist −1,437 Limit to Infinity of Binomial Coefficient over Power No edit summary
24 April 2024
- 02:5302:53, 24 April 2024 diff hist +2,608 N Principle of Open Induction for Real Numbers Created page with "== Theorem == Let $a < b$ be real numbers. Let $S$ be an open set of real numbers. Suppose that, for every $x \in \closedint a b$ such that: :$\hointr a x \subseteq S$ it also holds that: :$x \in S$ Then, $\closedint a b \subseteq S$. == Proof == {{AimForCont}} there exists some $x \in \closedint a b$ such that: :$x \notin S$ Let: :$T := \closedint a b \setminus S$ be the set o..."
22 April 2024
- 05:3705:37, 22 April 2024 diff hist 0 m Differentiation of Power Series No edit summary current
- 05:3705:37, 22 April 2024 diff hist 0 m Nth Derivative of Mth Power No edit summary current
25 March 2024
- 23:3723:37, 25 March 2024 diff hist +1,183 N Filter Containing Complements is Not Proper Created page with "== Theorem == Let $L = \struct {S, \lor, \land, \preceq}$ be a bounded lattice. Let $F \subseteq S$ be a filter on $L$. Suppose there exist $a, b \in F$ such that: :$b$ is a complement of $a$ Then, $F = S$. == Proof == By filter axiom $\paren 2$: :$\exists c \in F: c \preceq a \land c \preceq b$ By definition of Definition:Complement (Lattice..."
- 18:5118:51, 25 March 2024 diff hist +744 Definition talk:Limit Point/Filter No edit summary current
21 March 2024
- 21:0421:04, 21 March 2024 diff hist −1 m Riemann-Stieltjes Integral with Step Integrator No edit summary current
- 18:5118:51, 21 March 2024 diff hist +20 m Riemann-Stieltjes Integral with Step Integrator No edit summary
- 18:5018:50, 21 March 2024 diff hist +55 m Reduction of Riemann-Stieltjes Integral to Identity Integrator No edit summary current
- 18:5018:50, 21 March 2024 diff hist +3,987 Riemann-Stieltjes Integral with Step Integrator No edit summary
- 16:5716:57, 21 March 2024 diff hist +6,915 N Riemann-Stieltjes Integral with Step Integrator Created page with "== Theorem == Let $a < c < b$ be real numbers. Let $f$ be a real function that is bounded on $\closedint a b$. Let $\alpha$ be a real function on $\closedint a b$ such that: :$\forall x \in \hointr a c: \map \alpha x = \map \alpha a$ :$\forall x \in \hointl c b: \map \alpha x = \map \alpha b$ Suppose that: :Either $f$ is Definition:Left-C..."
- 05:1905:19, 21 March 2024 diff hist −104 Left-Hand and Right-Hand Differentiable Function is Continuous I extracted this theorem for something else, but it simplifies the logic here accidentally. current
- 05:1605:16, 21 March 2024 diff hist +1,566 N Continuous at Point iff Left-Continuous and Right-Continuous Created page with "== Theorem == Let $A \subseteq \R$ be an open set of real numbers. Let $f : A \to \R$ be a real function. Let $x_0 \in A$. Then: :$f$ is continuous at $x_0$ {{iff}}: :$f$ is both left-continuous and right-continuous at $x_0$ == Proof == === Necessary Condition === Su..." current
- 03:1303:13, 21 March 2024 diff hist 0 Book:Tom M. Apostol/Mathematical Analysis/Second Edition I checked both on my physical copy, as well as https://openlibrary.org/books/OL5291289M/Mathematical_Analysis and https://search.worldcat.org/title/827630. current
- 02:4302:43, 21 March 2024 diff hist +70 m Integration by Substitution/Riemann-Stieltjes Integral No edit summary
- 02:4202:42, 21 March 2024 diff hist +1 m Integration by Substitution/Riemann-Stieltjes Integral No edit summary
- 02:3802:38, 21 March 2024 diff hist +2,859 Integration by Substitution/Riemann-Stieltjes Integral Removed redirect to Integration by Substitution/Riemann-Stieltjes Integral/Increasing Tag: Removed redirect
- 02:2602:26, 21 March 2024 diff hist −55 m Integration by Substitution/Riemann-Stieltjes Integral/Decreasing No edit summary current
- 02:2502:25, 21 March 2024 diff hist −28 m Integration by Substitution/Riemann-Stieltjes Integral/Increasing No edit summary current
- 02:2402:24, 21 March 2024 diff hist +5,553 N Integration by Substitution/Riemann-Stieltjes Integral/Decreasing I don't see an easy way to derive this from the Increasing case.
- 01:3401:34, 21 March 2024 diff hist +79 N Integration by Substitution/Riemann-Stieltjes Integral CircuitCraft moved page Integration by Substitution/Riemann-Stieltjes Integral to Integration by Substitution/Riemann-Stieltjes Integral/Increasing: Special case of general theorem ($g$ is monotone). Tag: New redirect
- 01:3401:34, 21 March 2024 diff hist 0 m Integration by Substitution/Riemann-Stieltjes Integral/Increasing CircuitCraft moved page Integration by Substitution/Riemann-Stieltjes Integral to Integration by Substitution/Riemann-Stieltjes Integral/Increasing: Special case of general theorem ($g$ is monotone).
- 01:1801:18, 21 March 2024 diff hist +7,385 N Reduction of Riemann-Stieltjes Integral to Identity Integrator We don't have good coverage for one-sided derivatives, etc., which is what is meant here.
- 01:1501:15, 21 March 2024 diff hist +420 Sum of Riemann-Stieltjes Integrals on Adjacent Intervals Is this a better way to organize it?
- 01:0001:00, 21 March 2024 diff hist 0 m Definition:Riemann-Stieltjes Integral/General Limits of Integration Whoops, wrong one.
- 00:5900:59, 21 March 2024 diff hist +9 m Definition:Riemann-Stieltjes Integral/General Limits of Integration No edit summary
- 00:5900:59, 21 March 2024 diff hist +8 m Sum of Riemann-Stieltjes Integrals on Adjacent Intervals/Whole to Part No edit summary current
- 00:5800:58, 21 March 2024 diff hist 0 m Sum of Riemann-Stieltjes Integrals on Adjacent Intervals/Whole to Part CircuitCraft moved page Sum of Riemann-Stieltjes Integrals on Adjacent Intervals/Part to Sum of Riemann-Stieltjes Integrals on Adjacent Intervals/Whole to Part without leaving a redirect: More descriptive name
- 00:5800:58, 21 March 2024 diff hist +9 m Sum of Riemann-Stieltjes Integrals on Adjacent Intervals/Part to Whole No edit summary current
- 00:5700:57, 21 March 2024 diff hist +8 m Linear Combination of Riemann-Stieltjes Integrals/Integrator No edit summary current
- 00:5700:57, 21 March 2024 diff hist 0 m Sum of Riemann-Stieltjes Integrals on Adjacent Intervals/Part to Whole CircuitCraft moved page Sum of Riemann-Stieltjes Integrals on Adjacent Intervals/Whole to Sum of Riemann-Stieltjes Integrals on Adjacent Intervals/Part to Whole without leaving a redirect: Better naming convention.
20 March 2024
- 20:4120:41, 20 March 2024 diff hist +41 m Sum of Riemann-Stieltjes Integrals on Adjacent Intervals No edit summary
- 07:0907:09, 20 March 2024 diff hist +244 Talk:Sum of Riemann-Stieltjes Integrals on Adjacent Intervals No edit summary
- 07:0507:05, 20 March 2024 diff hist +154 N Talk:Sum of Riemann-Stieltjes Integrals on Adjacent Intervals Created page with "Subpages are transcluded lower down the page as lemmata. --~~~~"
- 07:0407:04, 20 March 2024 diff hist −297 m Sum of Riemann-Stieltjes Integrals on Adjacent Intervals Reverted edits by Prime.mover (talk) to last revision by CircuitCraft Tags: Rollback Reverted
- 06:0706:07, 20 March 2024 diff hist −1,051 Riemann-Stieltjes Integral by Norm of Subdivision No edit summary current
- 06:0506:05, 20 March 2024 diff hist +1,853 N Existence of Subdivision with Small Norm Created page with "== Theorem == Let $\closedint a b$ be a closed real interval. Let $\epsilon > 0$ be a positive real number. Then, there exists a finite subdivision $P$ of $\closedint a b$ such that: :$\norm P < \epsilon$ where $\norm P$ denotes the norm of $P$. == Proof == By the Axiom of Archimedes, choose $N \in \N$ such that:..." current
- 05:3805:38, 20 March 2024 diff hist −197 Riemann-Stieltjes Integral by Norm of Subdivision No edit summary
- 05:2905:29, 20 March 2024 diff hist +1,695 N Norm of Refinement is no Greater than Norm of Subdivision Created page with "== Theorem == Let $P$ be a finite subdivision of $\closedint a b$. Let $P'$ be a refinement of $P$. Then: :$\norm {P'} \le \norm P$ where $\norm P$ denotes the norm of $P$. == Proof == Let $P = \set {x_0, x_1, \dotsc, x_{n - 1}, x_n}$. Let $P' = \set {y_0, y_1, \dotsc, y_{m - 1}, y_m}$. By definition of Definition:Refinement of Finite Subdivision..." current
- 04:3304:33, 20 March 2024 diff hist −70 m Definition:Riemann-Stieltjes Integral Ended up changing the naming convention for that page. I'll put back the section when the full equivalence proof is complete.
- 04:2804:28, 20 March 2024 diff hist +46 m Riemann-Stieltjes Integral by Norm of Subdivision/Riemann Integral No edit summary current
- 04:2704:27, 20 March 2024 diff hist +41 m Riemann-Stieltjes Integral by Norm of Subdivision No edit summary
- 04:2704:27, 20 March 2024 diff hist +2,443 N Riemann-Stieltjes Integral by Norm of Subdivision/Riemann Integral Created page with "== Theorem == <onlyinclude> Let $f$ be a real function that is bounded on $\closedint a b$. Suppose $f$ is Riemann integrable on $\closedint a b$. Let $\iota$ be the identity mapping on $\closedint a b$. Then, $f$ is Riemann-Stieltjes integrable with respect to $\iota$ on $\closedint a b$ an..."
- 03:5703:57, 20 March 2024 diff hist +3,489 N Riemann-Stieltjes Integral by Norm of Subdivision Created page with "== Theorem == Let $f, \alpha$ be real functions that are bounded on $\closedint a b$. Suppose there exists some $A \in \R$ where, for every $\epsilon > 0$, there exists some $\delta_\epsilon > 0$ such that: :For every finite subdivision $P$ of $\closedint a b$, if the norm $\norm P < \delta_\epsilon$, then: ::For every Definiti..."
- 03:5703:57, 20 March 2024 diff hist −45 m Riemann-Stieltjes Integral of Constant Integrand No edit summary