# Tagged Questions

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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### Root Test: What happens if $\lim\limits_{n\rightarrow \infty} \sqrt[n]{|a_n|} =1$?

Root Test: For a series $\sum\limits_{n=1}^\infty a_n$, let $\ell = \lim\limits_{n\rightarrow \infty} \sqrt[n]{|a_n|}$. Series is convergent if $\ell <1$ and divergent if $\ell \geq 1$. I have ...
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### Computing $\delta$ as a function of $\epsilon$ for uniformly continuous functions

So a function is uniformly continuous if $\forall \epsilon > 0, \exists \delta >0$ s.t. $\forall x,y$, $|x-y|<\delta \rightarrow |f(x)-f(y)| < \epsilon$. Is there another category of ...
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### Finding a 2-chain whose boundary is $c - c_{1, n}$ for a 1-cube $c$ such that $c(0) = c(1)$ and a circle $c_{1, n}$.

Problem (Spivak, 4-24). Define $c_{1, n} : [0, 1] \to \mathbb{R}^2$ by $c_{1, n}(t) = (\cos 2\pi nt, \sin 2\pi nt)$. If $c$ is a singular 1-cube in $\mathbb{R}^2-\{ 0 \}$ with $c(0) = c(1)$, show ...
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### Is this set countable or uncountable? (Related to mean value theorem)

Let $f$ be a differentiable function on the real line. Consider any $h\in (0,1)$. By the mean value theorem there exists $d_h\in (0,h)$ such that $f'(d_h)=\frac{f(h)-f(0)}{h}$. Is the set $\{d_h\}$ ...
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### Construction of a measure space from some weird functional

Here is the complete problem I am trying to solve, but currently, I am just interested in proving that $\Sigma$ is a $\sigma$-algebra. Let $X$ be a set and $\phi: 2^X \to [0, \infty]$ be a ...
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### Smooth Approximations in $L^2((0,1))$

Let $L^2((0,1))$ be as usual the Lebesgue space of measurable complex-valued functions $f:(0,1) \rightarrow \mathbb{C}$ such that $\int |f(x)|^2 dx < \infty$. It is a well known fact (see e.g Lieb ...
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### A $C^1$ function in Orlicz Sobolev space

How to prove that thi functional is $C^1$: $$I(u)=\int_{\mathbb{R}^N} \Phi(|\nabla u|)+\Phi(|u|) dx-\int_{\mathbb{R}^N} F(u) dx$$ Where $\Phi$ is an N-function and $F(t)=\int_{0}^t f(s) ds$ where ...
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### Good, relatively short math textbooks? [closed]

Recently I've been trying to decide on some fun math summer reading on some areas of math which I have less experience with. I'm an undergrad studying mathematics with a focus in actuarial science, ...
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### Closure of $\frac {1} {n}$ [duplicate]

I have as a definition of the closure of a set $E$ in a metric space $(S,d)$ that $E^-$ is the intersection of all closed sets containing E (Elementary analysis the theory of calculus by Kenneth Ross)....
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### Why is sum $\cot^{-1}(n^2+n+1)$ equal to $\cot^{-1}\left(\frac1{m+1}\right)$?

Can u Please Explain This sum result : $$\sum_{n=0}^m \cot^{-1} (n^2+n+1) = \cot^{-1} \left(\frac1{m+1}\right)$$
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### $f \colon \mathbb{R}^n \to \mathbb{R}^n$ of class $C^1$ such that $||f(x) - f(y)|| \geq c||x-y||$ is a diffeomorphism.

Let $f \colon \mathbb{R}^n \to \mathbb{R}^n$ be a $C^1$ map. Suppose that exists $c>0$ such that $||f(x)-f(y)||\geq c||x-y||$ for all $x,y \in \mathbb{R}^n$. Prove that $f$ is a diffeomorphism. I ...
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### How prove that the range of an rectifiable curve has measure zero?

Let $f:[a,b] \rightarrow \mathbb{R}^n$ be a rectifiable continuous curve, show that $f[a,b]$ has content zero. If $f$ is continuous then is integrable so we have that $[a,b]\times f[a,b]$ has ...
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I want to calculate $\int_{\partial B(x,\epsilon)}\frac{1}{|x-y|^2}$ with $B(x,\epsilon)\in \mathbb{R}^3$ Time ago I saw a paper who said (if I am right) $\int_{\partial B(0,R)}\frac{1}{|x-y|^{n-1}}=... 3answers 64 views ### Finding$n$th derivative in an unusual way If$f(z) = \frac{e^{iz}}{z^2-1}$, then$f^{(4)}(z)$can be found by differentiating$f(z)$four times. I tried to use Cauchy's integral formula, but the integrand is not holomorphic at$z=0$, so we ... 2answers 29 views ###$f(x)=x\ln x-\frac{k}{x}$and$f(x_1)=f(x_2)=0\Rightarrowf'\left(\frac{x_1+x_2}{2}\right)\not=0$If$f(x)=x\ln x-\frac{k}{x}$. And$x_1$,$x_2$are two roots of$f(x)=0$. Then$f'\left(\frac{x_1+x_2}{2}\right)\not=0$First, I determine the range of$k$. Because$f=0$has two roots$\iffx^2\...
I have been trying to prove or disprove: Let $g$ be differentiable on the reals, have $g'(x)\geq 0$ except countably many values, then $g$ is non-decreasing. The main problem I face is that I don't ...