How to prove the sequence of function $\{f_n\}$ is Cauchy but not convergent?...
ProblemConsider the space $C[-1,1]$, together with the norm defined by $\|f\|_1 = \int_{-1}^1|f|d\lambda$ (where $\lambda$ is the Lebesgue measure). For each $n$ define a function...
View ArticleFlux through a paraboloid in the first quadrant
The question is absolutely easy but I am unsure where I go wrong haha. I am given a vector field $\mathbf{u} = (y,z,x)$ and I need to find the ourward flux $\iint\limits_{M} \textbf{u} \cdot...
View ArticleWhat is the meaning of differentiable?
Definition on my text book for differentiable is: for a point c, if$f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}$ , then f is differentiable at cI'm confused that $f'(c)$ has a very similar...
View ArticleConvergent series for fixed x
Prove that the series $ \sum_{n=0}^{\infty} \cos^2(x)\sin^{2n}(x)$ is convergent for fixed x in $(-\frac{\pi}{2},\frac{\pi}{2})$, and determine the sum. Prove, that the sum does not depend on x in...
View ArticleDid my teacher make a mistake? [closed]
I had to prove that the series $ \sum_{n=0}^{\infty} \cos^2(x)\sin^{2n}(x)$ is convergent for fixed x in $(-\frac{\pi}{2},\frac{\pi}{2})$, determine the sum, and prove, that the sum does not depend on...
View ArticleExistence and Uniqueness of the Median of Lipschitz Functions on $(X,d,\mu)$
Let $\varphi:(X,d,\mu)\to \Bbb R$ be a Lipschitz function, where $\mu$ is a probability measure on the metric space $(X,d)$. The median $m_\varphi$ of $\varphi$ is defined as the real number such...
View ArticleUniform continuity inequality check [closed]
This question is about how the person in a linked question (below), managed to derive a certain inequality. I present the linked question as well as my own derivation. I believe my question is a useful...
View ArticleDetermining Absolute Convergence of a Series
Question:Hello, I'm currently studying series in my calculus course and I've come across a problem that I'm having trouble with. The problem is to determine whether the following series is absolutely...
View ArticleProb. 3, Sec. 2.8 in Erwine Kreyszig's Introductory Functional Analysis with...
Let $C[-1,1]$ denote the normed space of all (real or complex-valued) functions defined and continuous on the closed interval $[-1,1]$ on the real line, with the norm given by$$\Vert x \Vert_{C[-1,1]}...
View ArticleIntegrate $1/x$ in the sense of distribution, with non-symmetric excluded...
I'm kind of confused with this question because of the boundary terms. for a test function $\phi$ define: $ u(\phi) = \lim_{\epsilon \rightarrow 0+} \int_{x\notin(-3\epsilon,5\epsilon)} \phi(x) x^{-1}...
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A simple question about the completness of $L^p$
Since $g<\infty$ a.e the series $$\sum_{i=1}^\infty (f_{n_{i+1}}-f_{n_i})$$ converges a.e in $X$.From this, why also the series $$f_{n_1}+\sum_{i=1}^\infty (f_{n_{i+1}}-f_{n_i})$$ converges a.e in...
View ArticleNotation $D^{m}$ in certain inequalities in $L^{p}$
In this wikipedia article, one can find the so-called "Gagliardo-Nirenberg inequality, which are of the form$${\displaystyle \|D^{j}u\|_{L^{p}(\mathbb {R} ^{n})}\leq C\|D^{m}u\|_{L^{r}(\mathbb {R}...
View ArticleCalculate integral $\int\limits_{0}^{\infty }xe^{-nx}\cos xdx$
Calculate integral$$\int\limits_{0}^{\infty }xe^{-nx}\cos xdx$$$$I=\int\limits_{0}^{\infty }xe^{-nx}\cos xdx=\Re \left \{ \int\limits_{0}^{\infty }xe^{-nx}e^{ix}dx \right \}=\Re \left \{...
View ArticleWhat function satisfies $f^{(p)}(0)=0$ for all $p$ except one value other...
What is an example of a function satisfying $$f^{(p)}(0)=0$$ for all $p$ except exactly one value other than the power function $x^n$?I can't think of any function and don't know if any exist either.
View ArticleLinear mapping from $\mathbb{R}^n$ to $\mathbb{R}$
Let $f:\mathbb{R}^n\to\mathbb{R}$ be the linear mapping.a) Is $f$ uniformly continuous on $\mathbb{R}^n$?b) Is $f$ differentiable on $\mathbb{R}^n$?c) Find the general form of $f$.My attempt for part...
View ArticleProve that if $\{a_{k}\}$ is a sequence of real numbers such that...
Prove that if $\{a_{k}\}$ is a sequence of real numbers such that$$\sum_{k=1}^{\infty} \frac{|a_{k}|}{k} = \infty$$and$$\sum_{n=1}^{\infty} \left( \sum_{k=2^{n-1}}^{2^n-1} k(a_k - a_{k+1})^2...
View ArticleUse Calculus results (not numeric integration methods) to approximate...
Consider$$\int_{-1}^2 e^{-x^2}dx$$How can I approximate its value using Calculus theory? That is, not using numerical methods such as the trapezoid rule taught in Numerical Mathematics...
View ArticleQuestion regarding sigma finite set proof
It’s a well known fact that, If $f$ is integrable, then $\{x\in X\mid f(x)\neq 0\}$ is sigma finite. Here is a proof : If $f$ is integrable, then $\{x\in X\mid f(x)\neq 0\}$ is sigma finiteNow, the...
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Prove limit as $x\to 0$ of $(1/x)\sin(1/x)$
Prove limit as $x\to 0$ of $(1/x)\sin(1/x)$See my working below, hopefully you find it legible.Proving it using epsilon-delta, turned it into a proof by contradiction.I assume the limit converges to...
View ArticlePositive integral of $fg$ everywhere for all $g \in C_C^0(\mathbb R^n )$...
Let $f$ be an locally integrable function on the measure space $(\mathbb R^n,S,\mu)$, with $\mu$ be a radon measure proof that\begin{align}\text{If }\int_{\mathbb R^n} f g\, d\mu \geq 0\text{ for all...
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