I want to show that $C^2$([0,1]) is not complete with respect to the $C^1$ norm.Recall that $||f||_{C^1}$ = $||f||_\infty$+$||f'||_\infty$.I can come up with counter-examples of why $C^1$[0,1] is not complete w.r.t the sup norm with $f_n=(x+\frac{1}{ ## Calculus (Limits) Doubt:$\theta\cfrac{\theta^3}{3} \sin \theta \theta$use to solve limit. December 1 Following is the question I've been trying to work on but can't get enough of it: $$\lim_{n\rightarrow \infty} \sin\left(\cfrac{n}{n^2+1^2}\right) + \sin\left(\cfrac{n}{n^2+2^2}\right) + \cdots + \sin\left({\cfrac{n}{n^2+n^2}}\right)$$ I'm required ## question on finding limit December 1 1 Let$x_n=\left(1-\frac13\right)^2\left(1-\frac16\right)^2\left(1-\frac1{10}\right)^2.....\left(1-\frac1{\frac{n(n+1)}2}\right)^2,n\ge2$then find$\lim_{n\rightarrow \infty}~ x_n$?while answering please mention the underlying concept too.Hint:Try rewr ## Calculus (Limits) Question December 1 Following is the question I've been trying to work on but can't get enough of it: $$\lim_{n\rightarrow \infty} \sin\left(\cfrac{n}{n^2+1^2}\right) + \sin\left(\cfrac{n}{n^2+2^2}\right) + \cdots + \sin\left({\cfrac{n}{n^2+n^2}}\right)$$ I'm required ## Removing impact of the change of a single term from the change in a weighted geometric mean December 1 I'm look at the the Dollar Index which according to wikipedia is a weighted geometric mean. What I want to do is to remove the impact of a single currency from the basket. Would it be correct to take the percentage change in the DXY and subtract out ## Understanding the notation$N/(N_{r}\bigcup N_{v})$in graph theory December 1 2 Currently I'm dealing with a graph problem but I don't understand one specific notation. What does the following mean:$$N/(N_{r}\bigcup N_{v})$$$N$,$N_{r}$,$N_{v}$are sets of nodes.$N$i ... ## Why does higher level mathematics more often than not use Greek lettering December 1 3 In high school, at least from what I've seen, mathematics courses never use Greek lettering in their description of concepts, with the notable exceptions of$\Sigma$for summations,$\Delta$for changes over time,$\pi$as$3.14159\ldots$,$\tau$in ## Is it okay to write$f(x)^2$December 1 1 As far as I understand we write$\cos^2 x$just to not mix up$(\cos x)^2$with$\cos(x^2)$. But it is difficult to associate$f(x)^2$with anything else but$(f(x))^2$. Is it correct to use such a notation?by convention $$\cos^2x=(\cos x)^2$$ if $$f ## Why do people use K to represent a complete graph December 1 2 Why do people use the letter "K", rather than "C", to represent a complete graph? Does it come from German "komplett"?My understanding is that Harary introduced the notation K_5 and K_{3,3} for the graphs appearing in Kur ## Why is the slope-intercept form of the equation of a line often written y=mx+b Why m instead of a December 1 1 After a quick google search, I read something about Conway suggesting the m having to do with "modulus" ... This seems odd to me, but perhaps there is some mathematical reason? I've heard of the uses of the word modulus in real/complex analysi ## Why not write \sqrt{3}2 December 1 4 Is it just for aesthetic purposes, or is there a deeper reason why we write 2\sqrt{3} and not \sqrt{3}2?The format \sqrt{3}2 is easliy confused with \sqrt{32}.I also suspect that many early typesetters would skip the overline, so that \sqrt{ ## Why is Fermat's spiral formula written as r^2=a^2\theta instead of r=a\sqrt{\theta} December 1 2 I'm reading Clifford A. Pickover's Math Book, in the Fermat's spiral page, it says the Fermat's spiral formula is r^2=a^2\theta, why isn't it written as r=\pm a\sqrt{\theta}? What's the problem in writing it that way?Aesthetics: polynomial equati ## Why do some people use +\infty instead of \infty December 1 6 Why do some people use +\infty instead of \infty? Because we usually denote minus infinity as -\infty, I think it is sufficient to denote plus infinity as \infty. Are there any reasons for this notation? I saw that many people majoring analys ## Why 1\frac{1}{2}\ne \frac{1}{2} December 1 1 Why mathematicians have chosen notation such that in algebra 1\frac{1}{2}=\frac{3}{2} but x\frac{y}{z}=\frac{xy}{z}, instead of x\frac{y}{z}=\frac{xz+y}{z}?$$1\frac12=1+\frac12$$and not$$1\cdot\frac12$$4 6 ## Why do we write f : X \rightarrow Y as opposed to f \in X \rightarrow Y. December 1 1 I've always been taught to write f : X \rightarrow Y as opposed to f \in X \rightarrow Y. This seems weird though, since X \rightarrow Y can be viewed as the set of all functions with source X and target Y, in which case the notation f \in ## Can every measure be normalized in order to be a probability measure December 1 1 Let \mu be a measure on (X,\mathcal{A}). Is it possible to normalize \mu in order to get a probability measure? My idea is to set$$ \mu'(A):=\mu(A)/\mu(X)~\forall~A\in\mathcal{A}. $$As pointed out in the comments, if \mu is a finite measure ## Restriction of probability measure December 1 Suppose that \Omega is a finite set and \Pr is a probability measure on it. Let A be a subset of \Omega. What is the correct notation for the "restriction" of \Pr to A, i.e. what's the correct notation for the probability measure \m ## Why do we use a single 'dash' for difference : - and a double 'dash' for sum: + December 1 Why do we use a single 'dash' for difference : - and a double 'dash' for sum: +?Just a shower thought: Who came up with this notation? It kind of makes it look like the difference is simpler than the sum, when intuitively, in my opinion, it shoul ## Prove polynomial and verification December 1 Suppose we have a polynomial f(x)=a_nx^n+...a_1x+a_0 where a_i is in real number, i is between 0 and n. Prove that f(z)=0 implies f(\overline z)=0Then let f(x)=x^4+4. Verify that f(1+i)=0. Then find the other three roots from 1+i ## Show that \omega^{2}\wedge\cdots\wedge\omega^{2} n times is equal to December 1 Consider \mathbb{R}^{2n} with coordenates x^{1},\cdots,x^{2n} and the following differential form of grade two$$\omega^{2}=dx^{1}\wedge dx^{n+1}+dx^{2}\wedge dx^{n+2}+\cdots+dx^{n}\wedge dx^{2n}$$Show that \omega^{2}\wedge\cdots\wedge\omega^{2 You Might Also Like • I am a beginner in working with statistics and I wanted to generate a bivariate Poisson distribution (X_1, ... • Recently, we investigated whether the expression in the central limit theorem converges to a random variable ... • Wolfram MathWorld states that$$ \sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} = \frac{ \pi \sqrt{3}}{18 ... • A monkey at a typewriter types each if the 26 letters of the alphabet exactly once, the order being random.A ... • Recently I've started to take interest in linear diophantine equations (they play a key role in a math puzzl ... • I know basic geometry and algebra, I know trigonometry but I don't understand it, I'm willing to learn math ... • I was reading about the Riemann zeta function in the region Re(Z) > 1, where it can be represented by the ... • Let$V(x,y)$represent a two-variable positive definite function, i.e.,$V(x,y):\mathbb{R}\times\mathbb{R}\t ...
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