Show that $f'$ is unbounded on $(0,1]$ but that $f$ is nevertheless uniformly...
The question is this. Let $f(x) = \sqrt{x}, \forall x \geq 0.$ Show that $f'$ is unbounded on $(0,1]$ but that $f$ is nevertheless uniformly continuous on $(0,1].$ Compare with Theorem 19.6.The theorem...
View ArticleLebesgue integral as a Summation like Riemann
I'm an IB student who currently just finished Math 31 HL (calculus grade 12). For an Internal assessment math paper I wanted to try and derive a formula that works like the Riemann summ but rather than...
View ArticleThe area of the area bounded by curve: $ \{ (x,y) \in \mathbb{R}^2 | x > 0, y...
Show that the set:$$ \{ (x,y) \in \mathbb{R}^2 | x > 0, y < 0, x^4 -xy^2 - y^3 = 0 \} \cup { (0,0) }$$is a closed curve. Calculate the area of the area bounded by this curve.Hint: substitute: $y...
View ArticleProving that $\iint f(x) g(x+y) dx dy \leq \int g^2(x) dx$
Let $D \subset \mathbb{R}^n$ be bounded, and $f,g: \mathbb{R}^n \rightarrow \mathbb{R}$. Furthermore let $f$ be nonnegative and such that $\int_D f dx= 1$.I would like to prove that$$\int_D \int_D...
View ArticleShow that the multivariable function f(x,g(x)) is differentiable
Let $W\subset U\subset \mathbb{R}^n, V\subset \mathbb{R}^m$ be open sets and $f:U\times V\rightarrow \mathbb{R}^m, g:W\rightarrow V$ differentiable functions. Show that $f(\vec{x},g(\vec{x})), \vec{x}...
View ArticleIf a function is $1$-periodic then it writes as a holomorphic function of...
I want to prove the following:Suppose that $g \in H(\mathbb{H})$ (where $\mathbb{H}$ is the upper half-plane) and $g(z+1) = g(z)$ for all $z \in \mathbb{H}$. Then there exists a function $G \in...
View ArticleBringing the limit inside the integral with asymptotic equilvance?
I was working on a limit problem and I think I found the solution to it, but I am not sure that what I did is right.I am working with the following sequence of functions$$ f_{n}(x) = n\,...
View ArticleGenerating functions of Wallis integrals
simple question, how would one go about proving that$$\sum_{n\ge 0}^{}W_{n}x^{n}=\int_{0}^{\frac{\pi}{2}}\frac{dt}{1-x\cos t}$$from this, it results with substituing $u= \tan(t/2)$ that$$\sum_{n\ge...
View ArticleShow that from a system of closed intervals covering an open interval it is...
Show thata) from a system of closed intervals covering a closed interval it is not always possible to choose a finite subsystem covering the interval;b) from a system of open intervals covering an open...
View ArticleProofs of AM-GM inequality
The arithmetic - geometric mean inequality states that$$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$I'm looking for some original proofs of this inequality. I can find the usual proofs...
View ArticleQuestion about proof of the Intermediate Value Theorem
I am still working through Lindstrom's real analysis book Spaces, and am trying to understand and fill in the details of his proof of the Intermediate Value Theorem. He states the theorem in a special...
View ArticleUnderstanding the proof of $L^p(X,\mathscr{A},\mu)$ is complete ($1\leq p
BackgroundI have some questions when reading the proof of $L^p(X,\mathscr{A},\mu)$ is complete for $1\leq p<+\infty$. The proof is proceeded by showing that each absolutely convergent series in...
View Article$\lim_{n\to+\infty}\int_{0}^{2\pi}\left(1+\frac{\sin(x)}{n}\right)^{\frac{n}{...
I've been trying to compute the following limit:$$\lim_{n\ \to\ \infty}\int_{0}^{2\pi}\left[1 + \frac{\sin\left(x\right)}{n}\right]^{n/x}\,{\rm d}x$$I am not completely sure, but when $x\in (0,2\pi)$...
View ArticleEvaluate $\lim_{n \to \infty} \int_0^\infty \frac{1+ \frac{x}{\sqrt{n}}...
Evaluate$$\lim_{n \to \infty} \int_0^\infty \frac{1+ \frac{x}{\sqrt{n}} e^{-x/n}}{(x+1)^2} \hspace{0.1cm} dx$$As this is a limit of integrals, surely we will be using a (monotone/dominated/bounded)...
View Articlea sequence of characteristic functions converge uniformly near t=0, then they...
I encountered a problem when reading A course in probability theory by Kailai Chung:If the sequence of ch.f.'s $\{f_n\}$ converges uniformly in a neighborhood of the origin, then $\{f_n\}$ is...
View ArticleCalculate...
$$\mbox{Calculate}\quad\sum_{n = 0}^{\infty}\int_{1/2}^{\infty}\left(1 - {\rm e}^{-t}\right)^{n}{\rm e}^{-t^{2}}{\rm d}t$$Basically I don't know where to start.I was thinking of using Tonelli Theorem...
View ArticleA new sum that equals $-\frac12 \ln (\frac{\pi}{2})$
(See EDIT)I found this result while working another problem in two slightly different directions and it kind of took me by...
View ArticleDirect proof of Bolzano-Weierstrass using Axiom of Completeness
The author of my intro analysis text has an exercise to give a proof of Bolzano-Weierstrass using axiom of completeness directly. Let $(a_n)$ be a bounded sequence, and define the set $$S=\{x \in...
View ArticleAverage power of an inharmonic "PM" (Phase Modulated) signal
Based on some experiments, I conjecture that the average power of an inharmonic "PM" (in the sense that's used in music, i.e. Phase Modulated) signal tends to be the same as that of the carrier, "in...
View ArticleFind all values of parameters a, b such that the limit of the function is finite
From my uni's textbook:Find all values of parameters $a$ and $b$ such that the following limit is a finite number:$$\lim_{x \to 0}\frac{\sin(ax)-\arcsin (bx)-3x}{x^{3}}$$The fact that $x$ is...
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