# 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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### Is there another simpler method to solve this elementary school math problem?

I am teaching an elementary student. He has a homework as follows. There are 16 students who use either bicycles or tricycles. The total number of wheels is 38. Find the number of students using ...
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### Is it morally right and pedagogically right to google answers to homework? [closed]

This is a soft question that I have been struggling with lately. My professor sets tough questions for homework (around 10 per week). The difficulty is such that if I attempt the questions entirely ...
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### How to prove that $\frac{(5m)!(5n)!}{(m!)(n!)(3m+n)!(3n+m)!}$ is a natural number?

How to prove that $$\frac{(5m)! \cdot (5n)!}{m! \cdot n! \cdot (3m+n)! \cdot (3n+m)!}$$ is a natural number $\forall m,n\in\mathbb N$ , $m\geqslant 1$ and $n\geqslant 1$? Thanks in advance.
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### Computing the Chern-Simons invariant of SO(3)

I am an undergraduate learning about gauge theory and I have been tasked with working through the two examples given on pages 65 and 66 of "Characteristic forms and geometric invariants" by Chern and ...
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### Surprising but simple group theory result on conjugacy classes

I have read that for any group $G$ of order $2m+1$ (odd) with $n$ conjugacy classes, it is always the case that $16$ divides the value $(2m+1)-n = |G|-n$. This seems to me like an astonishing ...
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### If $\sum a_n b_n <\infty$ for all $(b_n)\in \ell^2$ then $(a_n) \in \ell^2$

I'm trying to prove the following: If $(a_n)$ is a sequence of positive numbers such that $\sum_{n=1}^\infty a_n b_n<\infty$ for all sequences of positive numbers $(b_n)$ such that ...
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### The sum of irrationals is irrational?

If $x$ and $y$ are irrational, is $x + y$ irrational? Is $x - y$ irrational? Thanks for your help
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### If $a^2$ divides $b^2$, then $a$ divides $b$

Let $a$ and $b$ be positive integers. Prove that: If $a^2$ divides $b^2$, then $a$ divides $b$. Context: the lecturer wrote this up in my notes without proving it, but I can't seem to figure out ...
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### How to find the Galois group of a polynomial?

I've been learning about Galois theory recently on my own, and I've been trying to solve tests from my university. Even though I understand all the theorems, I seem to be having some trouble with the ...
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### Is there a continuous bijection from $\mathbb{R}$ to $\mathbb{R}^2$

I need a hint. The problem is: is there a continuous bijection from $\mathbb{R}$ to $\mathbb{R}^2$ I'm pretty sure that there aren't any, but so far I couldn't find the proof. My best idea so far is ...
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### Convergence of the series $\sum_{n=1}^\infty \frac{(\sin n)^n}{n}$.

Please determine whether the series $\displaystyle\sum_{n=1}^\infty \frac{(\sin n)^n}{n}$ converges. (Note: In Mathematica, the result tends to converge. Moreover, this is a problem mis-copied from ...
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### $\lim\limits_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite

How do I prove that $\displaystyle\lim_{n \to{+}\infty}{\sqrt[n]{n!}}$ is infinite?
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### Integral of $\frac{1}{(1+x^2)^2}$

I am in the middle of a problem and having trouble integrating the following integral: $$\int_{-1}^1\frac1{(1+x^2)^2}\mathrm dx$$ I tried doing partial fractions and got: $$1=A(1+x^2)+B(1+x^2)$$ I ...
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### Why is the ring of matrices over a field simple?

Denote by $M_{n \times n}(k)$ the ring of $n$ by $n$ matrices with coefficients in the field $k$. Then why does this ring not contain any two-sided ideal? Thanks for any clarification, and this is ...
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### Limit of series involving ratio of two factorials

$$\sum^{\infty}_{j=0} \frac{(j!)^2}{(2j)!} = \frac{2 \pi \sqrt{3}}{27}+\frac{4}{3}$$ The above series is in a homework sheet. We're not expected to find the limit, just prove its convergence. ...
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### Approximation to $\sqrt{2}$

I'm a first year Undergraduate student from India. Our professor is going to start a Real Analysis course in September and I was preparing for the initials. I tried and solved many problems, but this ...
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### Does the series $\sum\limits_{n=1}^{\infty}\frac{\sin(n-\sqrt{n^2+n})}{n}$ converge?

I'm just reviewing for my exam tomorow looking at old exams, unfortunately I don't have solutions. Here is a question I found : determine if the series converges or diverges. If it converges find ...
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### The union of a strictly increasing sequence of $\sigma$-algebras is not a $\sigma$-algebra

The union of a sequence of $\sigma$-algebras need not be a $\sigma$-algebra, but how do I prove the stronger statement below? Let $\mathcal{F}_n$ be a sequence of $\sigma$-algebras. If the ...
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### Easy proof, that $\rm e\notin \mathbb Q$

$\def\e{{\rm e}}$ I recently had the task to explain the proof that $\e$ is irrational as a presentation to my classmates. To prepare this presentation, the teacher gave me a script with a proof that ...
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### Seeking Fourier series solution on Laplace equation…still looking, am I on track?

Okay, I've been working at this a couple of days now, I will try to give relevant details but will omit some intermediate steps. The problem as given says: Consider the BVP for $u=u(x,y)$: ...
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### How to show $\det(AB) =\det(A)\det(B)$

Given two square matrices $A$ and $B$. How do you show $\det(AB) = \det(A)\det(B)$ where $\det(\cdot)$ is determinant of the matrix
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### Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.

Why is $$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$ Can we generalize it to any exponent $x \in \Bbb R$? This is to say, is $$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$ This is ...
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### Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$

Show:$$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \frac{1}{e}$$ So I can expand the numerator by geometric mean. Letting $C_{n}=\left(\ln(a_{1})+...+\ln(a_{n})\right)/n$. Let the numerator be called ...
### How to prove that $\frac{(mn)!}{m!(n!)^m}$ is an integer?
$\forall m,n\in\mathbb Z$ , $m\ge1$ and $n\ge1$ how to prove that $$\frac{(mn)!}{m!(n!)^m}$$ is an integer? Thanks in advance.