Study the convergence of the following series: $\sum_{n=0}^\infty \frac{...
I want to study the convergence of the following series: $\sum_{n=0}^\infty \frac{ n+x^{2n}}{n^3+1}$.Applying the necessary condition, I deduce that the series cannot converge for $x^2 > 1$. In the...
View ArticleRudin's RCA Theorem 7.21
Theorem 7.21 states: If $f:[a,b]\to\mathbb{R}$ is differentiable at every point of $[a,b]$ and $f'\in L^1$ on $[a,b]$, then $f(x)-f(a)=\int_a^x f'(t)dt$ for all $x\in[a,b]$.From a very early theorem,...
View ArticleA very interesting function equation $f(a,b)=f(a,c)+f(c,b)$ implies...
Function equation: $f(a,b)=f(a,c)+f(c,b)$ for all positive reals $a>c>b\geq 0$.My solution:$f(a,c)=f(a,b)-f(c,b)$,Let $b=0$,$f(a,c)=f(a,0)-f(c,0)$.Define $g(a)=f(a,0)$. We get the answer.It seems...
View ArticleWriting the definition of Upper Bound
Let X be an ordered set. Let $ S \subset X.$ An element $ u \in X$ is said to be an upper bound for $S$ if $s \leq u$ for all $ s \in S.$In first-order logic, how do I write the above definition? Is it...
View ArticleHow to find all 2nd degree polynomials solution to inequality in specific...
I'm trying to find the conditions on $a, b$, and $c$ so that a $2$nd-degree polynomial satisfies the following inequality when $x$ is in $[0,1]:$$0 < c + bx + ax^2 < 1$I found the following...
View ArticleShow that that $\lim_{n\to\infty}\sqrt[n]{\binom{2n}{n}} = 4$
I know that$$\lim_{n\to\infty}{{2n}\choose{n}}^\frac{1}{n} = 4$$but I have no Idea how to show that; I think it has something to do with reducing ${n}!$ to $n^n$ in the limit, but don't know how to get...
View ArticleA false proof of Dini theorem
I consider $C(X,\mathbb{R})$ the space of continuous functions from a compact metric space $X$ to $\mathbb{R}$ with the supremum norm. I would like to prove Dino’s theoremLet $(f_n)_n\subset...
View ArticleQuestion about $f$ of bounded variation
Def.The variation of $f$ on $[a, b]$ corresponding to a partition $P = \{x_0, x_1, \ldots, x_n\},$$V^P(f):=\sum_{i=1}^{n} |f(x_i) - f(x_{i-1})|.$Obviously, if $Q$ is a refinement of $P$, then $V^P(f)...
View ArticleLongtime beahviour of the heat kernel on the real line for bounded initial...
Let $u(t,x)$ be the fundemental solution to the heat equation $u_t = \frac{1}{2}u_{xx}$ with initial condition $u(0,\cdot)$. That is, $u(t,x) = \int_{\mathbb{R}}p_{t}(x-y)u(0,y)\mathrm{d}y$ where...
View ArticleImagining Rational Cauchy Sequences as Dancing Around a Real Number Instead...
I'm trying to build my intuition regarding the Cauchy-sequence construction of the reals.Essentially, do you think that it is more accurate to visualize a real number as being defined by sequences of...
View ArticleIf $f'$ is periodic , are $f , f'$ uniformly continuous?
$f: ℝ \rightarrow ℝ $ is such that $f'$ exist and $f'$ is periodic. Are $f ,f'$ uniformly continuous ?Attempt :- I was thinking along a counter example . What if $f'(x)=\tan x$ but then $f(x)=\ln (|sec...
View ArticleGaussian pdf unbounded variation on $\mathbb{R}$？
Let a real-valued function $f(x)$ defined on $\mathbb{R}$.For a bounded interval $[a,b]\subset \mathbb{R},$ taking a partition $\mathcal{P} =\{x_0, x_1, \ldots, x_n\},$ where $a\le...
View ArticleAbout the $n$th derivative of the Riemann zeta function on positive even...
I know there exist a formula for the Riemann zeta function on positive even integers involving Bernoulli numbers.Do there exist any closed form for the $n$th derivative of the Riemann zeta function on...
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Prove that the light ray sent from one focus of ellipse mirror end up in another
I have trouble proving the following problem from Zorich Mathematical Analysis ISo we have previously reached that the equation for the tangent line through $(x_0,y_0)$，we have the equation of the...
View ArticlePistol duel problem [closed]
Taes, Chinmay and Praveesh agree to fight a pistol duel under the following unusual conditions. After drawing lots to determine who fires first, second and third, they take their places at the corners...
View ArticleCentral limit theorem with changing bounds
Suppose $\{X_n\}_{n\in\mathbb{N}}$ is a sequence of iid random variables with $X_n \sim \text{Beta}(1/2, 1)$. I would like to upper bound the quantity$$ T = \lim_{d \to \infty}...
View ArticleSchwartz theorem
Can i relax the conditions on f of the Schwartz theorem in real analysis?That is: if $f: \Omega \to R^2$, s.t $f \in C^2(\Omega)$, then i can change the order of derivatives and the result is the...
View ArticleGradients of two functions with the same level sets are parallel for all...
Is the intuition behind this statement, based on : 1. The definition that a gradient is perpendicular to the level curves2. Since the level sets are the same for both functions, the corresponding...
View ArticleStronger than Lipschitz continuity on metric space
Let $(M, d)$ be a metric space (which can be compact and/or connected if needed).Is there a natural function class (of $M\rightarrow \mathbb{R}$-functions) that is strictly included in the Lipschitz...
View ArticleWhat is the need of induction in proving uniqueness of cardinality of sets?
In Analysis 1 by Terence Tao and here and basically every proof I read, I find that the proof is done by induction.But in the attached link the proof starts by stating that:$\mid A\mid =n$ and $\mid...
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