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1500 questions
65
votes
5 answers
Subgroups of finitely generated groups are not necessarily finitely generated
I was wondering this today, and my algebra professor didn't know the answer.
Are subgroups of finitely generated groups also finitely generated?
I suppose it is necessarily true for finitely generated abelian groups, but is it true in…
crasic
- 5,139
65
votes
6 answers
Is the power set of the natural numbers countable?
Some explanations:
A set S is countable if there exists an injective function $f$ from $S$ to the natural numbers ($f:S \rightarrow \mathbb{N}$).
$\{1,2,3,4\}, \mathbb{N},\mathbb{Z}, \mathbb{Q}$ are all countable.
$\mathbb{R}$ is not countable.
The…
Martin Thoma
- 10,157
65
votes
7 answers
Why does the volume of the unit sphere go to zero?
The volume of a $d$ dimensional hypersphere of radius $r$ is given by:
$$V(r,d)=\frac{(\pi r^2)^{d/2}}{\Gamma\left(\frac{d}{2}+1\right)}$$
What intrigues me about this, is that $V\to 0$ as $d\to\infty$ for any fixed $r$. How can this be? For fixed…
probabilityislogic
- 1,055
65
votes
19 answers
How do I convince my students that the choice of variable of integration is irrelevant?
I will be TA this semester for the second course on Calculus, which contains the definite integral.
I have thought this since the time I took this course, so how do I convince my students that for a definite integral
$$\int_a^b f(x)\ dx=\int_a^b…
ireallydonknow
- 1,435
65
votes
7 answers
How to prove that a $3 \times 3$ Magic Square must have $5$ in its middle cell?
A Magic Square of order $n$ is an arrangement of $n^2$ numbers, usually distinct integers, in a square, such that the $n$ numbers in all rows, all columns, and both diagonals sum to the same constant.
How to prove that a normal $3\times 3$ magic…
Archisman Panigrahi
- 2,688
65
votes
2 answers
Does $\sum _{n=1}^{\infty } \frac{\sin(\text{ln}(n))}{n}$ converge?
Does $\sum _{n=1}^{\infty } \dfrac{\sin(\text{ln}(n))}{n}$ converge?
My hypothesis is that it doesn't , but I don't know how to prove it. $ζ(1+i)$ does not converge but it doesn't solve problem here.
Darius
- 1,374
- 11
- 17
65
votes
16 answers
Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction
How can I prove that
$$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$
for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction.
Thanks
Steve
- 419
65
votes
1 answer
Are there simple examples of Riemannian manifolds with zero curvature and nonzero torsion
I am trying to grasp the Riemann curvature tensor, the torsion tensor and their relationship.
In particular, I'm interested in necessary and sufficient conditions for local isometry with Euclidean space (I'm talking about isometry of an open set -…
Selene Routley
- 2,735
65
votes
7 answers
How can I show that $\sqrt{1+\sqrt{2+\sqrt{3+\sqrt\ldots}}}$ exists?
I would like to investigate the convergence of
$$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4+\sqrt\ldots}}}}$$
Or more precisely, let $$\begin{align}
a_1 & = \sqrt 1\\
a_2 & = \sqrt{1+\sqrt2}\\
a_3 & = \sqrt{1+\sqrt{2+\sqrt 3}}\\
a_4 & =…
MJD
- 67,568
- 43
- 308
- 617
65
votes
1 answer
Why are asymptotically one half of the integer compositions gap-free?
Question summary
The number of gap-free compositions of $n$ can already for quite small $n$ be very well approximated by the total number of compositions of $n$ divided by $2$. This question seeks to understand why.
The details
A composition of an…
Daniel R
- 3,250
65
votes
13 answers
Is there an equation to describe regular polygons?
For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides required?
Using the Wolfram Alpha site, this input…
Vincent Tan
- 753
65
votes
2 answers
When are nonintersecting finite degree field extensions linearly disjoint?
Let $F$ be a field, and let $K,L$ be finite degree field extensions of $F$ inside a common algebraic closure. Consider the following two properties:
(i) $K$ and $L$ are linearly disjoint over $F$: the natural map $K \otimes_F L \hookrightarrow KL$…
Pete L. Clark
- 100,402
65
votes
15 answers
Why can't you flatten a sphere?
It's a well-known fact that you can't flatten a sphere without tearing or deforming it. How can I explain why this is so to a 10 year old?
As soon as an explanation starts using terms like "Gaussian curvature", it's going too far for the audience…
Joe
- 1,263
- 2
- 10
- 21
65
votes
12 answers
Are there mathematical concepts that exist in dimension $4$, but not in dimension $3$?
Are there mathematical concepts that exist in the fourth dimension, but not in the third dimension? Of course, mathematical concepts include geometrical concepts, but I don't mean to say geometrical concept exclusively. I am not a mathematician and…
tefisjb
- 1
65
votes
6 answers
Limits: How to evaluate $\lim\limits_{x\rightarrow \infty}\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\cdots+a_{0}}-x$
This is being asked in an effort to cut down on duplicates, see here: Coping with abstract duplicate questions, and here: List of abstract duplicates.
What methods can be used to evaluate the limit $$\lim_{x\rightarrow\infty}…
Eric Naslund
- 73,551