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

Understanding the notation $N/(N_{r}\bigcup N_{v})$ in graph theory December 1    2

Understanding the notation $N/(N_{r}\bigcup N_{v})$ in graph theory
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 ...

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 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 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

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

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 $f(\overline z)=0$Then let $f(x)=x^4+4$. Verify that $f(1+i)=0$. Then find the other three roots from $1+i$

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