# Tagged Questions

Homework questions are welcome as long as they are asked honestly, explain the problem, and show sufficient effort. Please do not use this as the only tag for a question. For the answers on homework questions, helpful hints or instructions are preferred to a complete solution. Please do not add ...

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### Evaluation of the limit $\lim\limits_{n \to \infty } \frac1{\sqrt n}\left(1 + \frac1{\sqrt 2 }+\frac1{\sqrt 3 }+\cdots+\frac1{\sqrt n } \right)$

Evaluate the limit : $$\lim_{n \to \infty } {1 \over {\sqrt n }}\left( {1 + {1 \over {\sqrt 2 }} + {1 \over {\sqrt 3 }} + \cdots + {1 \over {\sqrt n }}} \right)$$ I can use the sandwich principle, ...
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### If $x^3 =x$ then $6x=0$ in a ring

Let $R$ be a ring with unity where $$x^3=x,\;\;\; \forall x \in R$$ How do I prove that $$x+x+x+x+x+x=0$$
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### A matrix satisfying $AB-BA=B$

If $A$ and $B$ are two matrices of $\mathcal{M}_n(\mathbb{R}$) such that $$AB-BA=B$$ how can we prove that $B$ isn't invertible? my attempt: I found that $\mathrm{tr}(B)=0$ but I know that this is ...
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### What's wrong with these equations? [duplicate]

My friend Boris (Boryan) gave me a task, and completely refuses to give the answer what's wrong here. $$x^2=\overbrace{x+\cdots+x} ^{x\text{ times}}$$ $$(x^2)'=(x+\cdots+x)'$$ $$2x=1+\cdots+1$$ ...
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### Prove/Disprove that if two sets have the same power set then they are the same set

I am really sure that if two sets have the same power set, then they are the same set. I just am wondering how does one exactly go about proving/showing this? I'm usually wrong, so if anyone can show ...
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### Basis for $\mathbb{Z}^2$

Let $x = (a, b), y = (c, d) \in \mathbb{Z}^2$. What is the condition on $a, b, c, d$ so that ${x, y}$ is a basis? My answer: $ad\neq bc$ and $gcd(a, c) = gcd(b, d) = 1$. The first condition ...
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### Accumulation points of accumulation points of accumulation points

Let $A'$ denote the set of accumulation points of $A$. Find a subset $A$ of $\Bbb R^2$ such that $A, A', A'', A'''$ are all distinct. I can find a set $A$ such that $A$ and $A'$ are distinct, but not ...
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### Why is a finite integral domain always field?

This is how I'm approaching it: let $R$ be a finite integral domain and I'm trying to show every element in $R$ has an inverse: let $R-\{0\}=\{x_1,x_2,\ldots,x_k\}$, then as $R$ is closed under ...
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### Show determinant of matrix is non-zero

I have $a,b,c\in\mathbb{Q}$ not all zero. ($a^2+b^2+c^2\ne 0$), I want to show that the following determinant is then non-zero. I failed to arrive at an appropriate form of the polynomial. Help ...
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### Show $\lim \limits_{n \to \infty} \frac{a_{n+1}}{a_n} = \|f\|_{\infty}$ for $f \in L^{\infty}$

I have a question that I need help with getting started (possibly I would be back for more help). I have a measure space $(X,A,\mu)$ that is finite, and $f \in L^{\infty}(\mu)$. Also, defined is ...
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### Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$

As the title says, I'm wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem ...
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### Proof that there are infinitely many prime numbers starting with a given digit string

To prove the following fact: given any sequence of digits in any base, eg 314159265358979323 base 10, there are infinitely many primes that start with these digits,eg when expressed in decimal they ...
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### Showing $\gcd(n^3 + 1, n^2 + 2) = 1$, $3$, or $9$

Given that n is a positive integer show that $\gcd(n^3 + 1, n^2 + 2) = 1$, $3$, or $9$. I'm thinking that I should be using the property of gcd that says if a and b are integers then gcd(a,b) = ...
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### Suggestion for Computing an Integral

Let $$A=\left\{(x,y,z)\in \mathbb R^3:\dfrac{x^2}{2}+\dfrac{y^4}{4}+\dfrac{z^6}{6}\leq1\right\}.$$ Then I want to compute the following integral: ...
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### No simple group of order 300

So Ive been trying to prove that there's no simple group of order 300. This is what I did and I was wondering if it was enough. Let |$G$|=2$^2$ . 3 . 5$^2$. Suppose $G$ is simple. Then there would be ...
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### does $\intop_{1}^{\infty}x\sin(x^{3})dx$ really converge?

I'm trying to find a continuous function $f(x)$ on $[0,\infty)$ such that: $\intop_{1}^{\infty}f(x)dx$ converges while $f(x)$ isn't bounded. I came up with $f(x)=x\sin(x^{3})dx$, as a function ...
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### How to prove Lagrange trigonometric identity [duplicate]

I would to prove that $$1+\cos \theta+\cos 2\theta+\ldots+\cos n\theta =\displaystyle\frac{1}{2}+ \frac{\sin\left[(2n+1)\frac{\theta}{2}\right]}{2\sin\left(\frac{\theta}{2}\right)}$$ given that ...
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### Finding $\lim_{x \to 0}\ \frac{\sin(\cos(x))}{\sec(x)}$

The problem is to find: $\lim_{x \to 0}\ \dfrac{\sin(\cos(x))}{\sec(x)}$ I rewrite the equation as follows: $\lim_{x \to 0}\ \dfrac{\sin(\cos(x))}{\dfrac{1}{\cos(x)}}$ And multiply by ...
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The following is an Olympiad Competition question, so I expect it to have a pretty solution: For a positive integer $d$, define the sequence: \begin{align} a_0 &= 1\\ a_n &= ...
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### Isomorphism for Infinite Graphs.

Suppose that $G$ and $H$ are infinite graphs and that $G$ is isomorphic to a subgraph of $H$ and $H$ is isomorphic to a subgraph of $G$. Must $G$ and $H$ be isomorphic? I've only just started on ...
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### Prove that a UFD $R$ is a PID if and only if every nonzero prime ideal in $R$ is maximal

Prove that a UFD $R$ is a PID if and only if every nonzero prime ideal in $R$ is maximal. The forward direction is standard, and the reverse direction is giving me trouble. In particular, I can prove ...
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### Does $n \mid 2^{2^n+1}+1$ imply $n \mid 2^{2^{2^n+1}+1}+1$?

There are two ways to try to prove this. One is in the title, the other is its de Morgan counterpart: $n \nmid 2^{2^{2^n+1}+1}+1 \implies n \nmid 2^{2^n+1}+1$. Disproving it requires only one example ...
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### Random graph probability lemma

I'm trying to prove a fiddly lemma for homework, but getting absolutely nowhere with it. Here, $G_{n,p}$ and $G_{n,m}$ represent, respectively, random graphs on $n$ vertices where the number of edges ...
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### Minkowski Inequality for $p \le 1$

I've been trying to prove the concavity of a particular function which I reduced to proving the reverse Minkowski Inequality for $p \le 1$, $p \ne 0$ for arguments in $\mathbb{R}^{n}_{+}$. That is, ...
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### Polynomials with rational zeros

Find all polynomials $F(x)={a_n}{x^n}+\cdots+{a_1}x+a_0$ satisfying $a_n \neq0$; $(a_0, a_1, a_2, \ldots ,a_n)$ is a permutation of $(0, 1, 2 ... n)$; all zeros of $F(x)$ are rational.
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### Help me prove $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$

Please help me prove this Leibniz equation: $\sqrt{1+i\sqrt 3}+\sqrt{1-i\sqrt 3}=\sqrt 6$. Thanks!
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### Solving $5^n > 4,000,000$ without a calculator

If $n$ is an integer and $5^n > 4,000,000.$ What is the least possible value of $n$? (answer: $10$) How could I find the value of $n$ without using a calculator ?
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### Help with summing a power series

I'd like to determine the function corresponding to the following power series: $$x + \sum_{n=1}^\infty (-1)^n\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots2n} \frac{x^{2n+1}}{2n+1},$$ where ...
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### How to evaluate the series $1+\frac34+\frac{3\cdot5}{4\cdot8}+\frac{3\cdot5\cdot7}{4\cdot8\cdot12}+\cdots$

How can I evaluate the following series: $$1+\frac{3}{4}+\frac{3\cdot 5}{4\cdot 8}+\frac{3\cdot 5\cdot 7}{4\cdot 8\cdot 12}+\frac{3\cdot 5\cdot 7\cdot 9}{4\cdot 8\cdot 12\cdot 16}+\cdots$$ In one ...
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### Is $\mathbb{Q}/\mathbb{Z}$ isomorphic to $\mathbb{Q}$?

Is $\mathbb{Q}/\mathbb{Z}$ isomorphic to $\mathbb{Q}$? My guess is no. Does the first isomorphism theorem have anything to do with this? Any hints appreciated, thanks.
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### Factor $(a^2+2a)^2-2(a^2+2a)-3$ completely

I have this question that asks to factor this expression completely: $$(a^2+2a)^2-2(a^2+2a)-3$$ My working out: $$a^4+4a^3+4a^2-2a^2-4a-3$$ $$=a^4+4a^3+2a^2-4a-3$$ $$=a^2(a^2+4a-2)-4a-3$$ I am ...
### How to calculate $\int_{-a}^{a} \sqrt{a^2-x^2}\ln(\sqrt{a^2-x^2})\mathrm{dx}$
Well,this is a homework problem. I need to calculate the differential entropy of random variable $X\sim f(x)=\sqrt{a^2-x^2},\quad -a<x<a$ and $0$ otherwise. Just how to calculate  ...
### I want to know why $\omega \neq \omega+1$.
In Kunen's book, Set Theory,chapter I.7, he said: $1+\omega=\omega \neq \omega+1$. I want to know why $\omega \neq \omega+1$.