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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11
votes
4answers
1k views

How to solve this sequence $165,195,255,285,345,x$

This is a question appeared in a competitive exam. The question is: Find the unknown term in $165,195,255,285,345,x$ 1)375 $\ \ \ \ \ \ \ \ $ 2)420 3)435 $\ \ \ \ \ \ \ ...
11
votes
9answers
440 views

If $R$ and $S$ are fields, either prove or disprove that $R\times S$ is a field

That's the question from my homework. I am thinking $R\times S$ is not a field, but I'm not sure. I understand the definition of a field, but I am not sure how to proceed.
11
votes
6answers
719 views

Help me evaluate $\int_0^1 \frac{\log(x+1)}{1+x^2} dx$

I need to evaluate this integral: $\int_0^1 \frac{\log(x+1)}{1+x^2} dx$. I've tried $t=\log(x+1)$, $t=x+1$, but to no avail. I've noticed that: $\int_0^1 \frac{\log(x+1)}{1+x^2} dx = ...
11
votes
7answers
559 views

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, ...
11
votes
3answers
510 views

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 $$x+x+x+x+x+x=0$$
11
votes
6answers
752 views

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 ...
11
votes
7answers
1k views

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. $$x^2=\overbrace{x+\cdots+x} ^{x\text{ times}}$$ $$(x^2)'=(x+\cdots+x)'$$ $$2x=1+\cdots+1$$ ...
11
votes
3answers
4k views

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 ...
11
votes
3answers
758 views

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 ...
11
votes
3answers
259 views

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 ...
11
votes
6answers
2k views

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 ...
11
votes
5answers
468 views

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 ...
11
votes
4answers
595 views

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 ...
11
votes
5answers
2k views

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 ...
11
votes
4answers
943 views

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 ...
11
votes
4answers
1k views

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) = ...
11
votes
3answers
224 views

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: ...
11
votes
3answers
1k views

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 ...
11
votes
5answers
237 views

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 ...
11
votes
3answers
3k views

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 ...
11
votes
2answers
413 views

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 ...
11
votes
3answers
339 views

Collatz-ish Olympiad Problem

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 &= ...
11
votes
2answers
125 views

Prove that in a ring with $x^3 = x$, $x+x+x+x+x+x=0$.

This was an excercise on a course on abstract algebra at the University of Groningen. I have been working on this for ages, but I can't seem to figure it out. Problem Let $R$ be a ring with $\forall ...
11
votes
5answers
348 views

Prove that the sequence$ c_1 = 1$, $c_{n+1} = 4/(1 + 5c_n) $ , $ n \geq 1$ is convergent and find its limit

Prove that the sequence $c_{1} = 1$, $c_{(n+1)}= 4/(1 + 5c_{n})$ , $n \geq 1$ is convergent and find its limit. Ok so up to now I've worked out a couple of things. $c_1 = 1$ $c_2 = 2/3$ $c_3 = ...
11
votes
1answer
84 views

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 ...
11
votes
2answers
2k views

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 ...
11
votes
1answer
182 views

Calculate Asymptotics of Integral?

Let $f$ be a continuous function on $[0,1]$. How do I calculate the asymptotics, as $n\rightarrow\infty$, of $\displaystyle \int_{[0,1]^n}f\left(\frac{x_1+...+x_n}{n}\right)\text d x_1...\text d ...
11
votes
1answer
984 views

How to prove that a bounded linear operator is compact?

I encountered a homework problem that says: If $A$ is a bounded linear operator from $X$ to $Y$. And $K$ is a compact operator from $X$ to $Y$, where $X$ and $Y$ are both Banach spaces, and ...
11
votes
1answer
844 views

Spivak's “Differential Geometry” Volume 1, Chapter 1 ,Problem #20 part (b)

Problem 20 part (b) of Chapter 1 asks us to show that the infinite-holed torus is homeomorphic to the "infinite jail cell window." His hint helped me to get started (I think). (I apologize for not ...
11
votes
1answer
636 views

Exercise 6.5 in Humphrey's Book on Lie Algebras

I am trying to solve Exercise 6.5 part 4 in James Humphreys' Introduction to Lie Algebras and Representation Theory. I added the (homework) tag because my question is about an exercise, but this is ...
11
votes
3answers
2k views

how to prove DEF is an equilateral triangle?

ABC is an equilateral triangle,and AD = BE = CF,Prove DEF is an equilateral triangle.
11
votes
1answer
224 views

Class equation of subgroup of $SL(4,\mathbb{F}_2)$

Can you point me toward a computation-light derivation of the class equation of the subgroup of $SL(4,\mathbb{F}_2)$ consisting of upper-triangular matrices with 1's on the main diagonal? The ...
11
votes
1answer
517 views

Limit of a decreasing sequence of outer measures is an outer measure?

The problem given to me on my homework is: Prove that the limit of a decreasing family of outer measures is an outer measure. Doing out the "obvious" approach, we quickly reach the problem of ...
11
votes
1answer
488 views

Find the sum $\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + …+ \frac{1}{\sqrt{99}+\sqrt{100}}$

I would like to check I have this correct Find the sum $$\frac{1}{\sqrt{1}+\sqrt{2}} + \frac{1}{\sqrt{2}+\sqrt{3}} + ...+ \frac{1}{\sqrt{99}+\sqrt{100}}$$ Hint: rationalise the denominators ...
11
votes
1answer
691 views

Galois closure of a $p$-extension is also a $p$-extension

I'm working on a problem in Dummit & Foote and I'm quite stumped. The problem reads: Let $p$ be a prime and let $F$ be a field. Let $K$ be a Galois extension of $F$ whose Galois group is a ...
11
votes
1answer
817 views

Converse of the Weierstrass $M$-Test?

I was assigned a few problems in my Honors Calculus II class, and one of them was kind of interesting to do: Suppose that $f_{n}$ are nonnegative bounded functions on $A$ and let $M_{n} = \sup ...
11
votes
2answers
589 views

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 ...
11
votes
1answer
323 views

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 ...
11
votes
1answer
814 views

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, ...
11
votes
0answers
205 views

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.
11
votes
3answers
507 views

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!
10
votes
11answers
3k views

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 ?
10
votes
6answers
630 views

Given $a_{1}=1, \ a_{n+1}=a_{n}+\frac{1}{a_{n}}$, find $\lim \limits_{n\to\infty}\frac{a_{n}}{n}$

I started by showing that $1\leq a_{n} \leq n$ (by induction) and then $\frac{1}{n}\leq \frac{a_{n}}{n} \leq 1$ which doesn't really get me anywhere. On a different path I showed that $a_{n} \to ...
10
votes
5answers
708 views

Solve equations $\sqrt{t +9} - \sqrt{t} = 1$

Solve equation: $\sqrt{t +9} - \sqrt{t} = 1$ I moved - √t to the left side of the equation $\sqrt{t +9} = 1 -\sqrt{t}$ I squared both sides $(\sqrt{t+9})^2 = (1)^2 (\sqrt{t})^2$ Then I got $t + 9 ...
10
votes
4answers
917 views

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 ...
10
votes
2answers
453 views

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 ...
10
votes
3answers
810 views

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.
10
votes
5answers
650 views

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 ...
10
votes
5answers
434 views

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 $$ ...
10
votes
4answers
708 views

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 $\omega \neq \omega+1$.