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1500 questions
67
votes
2 answers

Continuous functions do not necessarily map closed sets to closed sets

I found this comment in my lecture notes, and it struck me because up until now I simply assumed that continuous functions map closed sets to closed sets. What are some insightful examples of continuous functions that map closed sets to non-closed…
Aaa
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67
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17 answers

Why does the derivative of sine only work for radians?

I'm still struggling to understand why the derivative of sine only works for radians. I had always thought that radians and degrees were both arbitrary units of measurement, and just now I'm discovering that I've been wrong all along! I'm guessing…
Kyle Delaney
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67
votes
1 answer

Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ Also, a few days ago, a friend of mine taught me that…
mathlove
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67
votes
6 answers

Foundation for analysis without axiom of choice?

Let's say I consider the Banach–Tarski paradox unacceptable, meaning that I would rather do all my mathematics without using the axiom of choice. As my foundation, I would presumably have to use ZF, ZF plus other axioms, or an approach in which sets…
user13618
66
votes
2 answers

Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $

I've found here the following integral. $$I = \int_{0}^{1}\sin{(\pi (1-x))}x^x(1-x)^{1-x}\,dx=\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx=\frac{\pi e}{24}$$ I've never seen it before and I also didn't find the evaluation on math.se. How could we…
user153012
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66
votes
2 answers

Evaluating $\int_{0}^{1}\cdots\int_{0}^{1}\bigl\{\frac{1}{x_{1}\cdots x_{n}}\bigr\}^{2}\:\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}$

Here is my source of inspiration for this question. I suggest to evaluate the following new one. $$ I_{n}:= \int_0^1 \! \cdots \! \int_0^1 \left\{\frac{1}{x_1x_2 \cdots x_n}\right\}^{2} \:\mathrm{d}x_1\,\mathrm{d}\,x_2 \cdots \mathrm{d}x_n…
Olivier Oloa
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66
votes
4 answers

What lies beyond the Sedenions

In the construction of types of numbers, we have the following sequence: $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$ or: $$2^0 \mathrm{-ions} \subset 2^1 \mathrm{-ions} \subset 2^2 \mathrm{-ions}…
Willem Noorduin
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66
votes
3 answers

What does "∈" mean?

I have started seeing the "∈" symbol in math. What exactly does it mean? I have tried googling it but google takes the symbol out of the search.
Locke
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66
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3 answers

Is $ \sum\limits_{n=1}^\infty \frac{|\sin n|^n}n$ convergent？

Is the series $$ \sum_{n=1}^\infty \frac{|\sin n|^n}n\tag{1}$$ convergent？ If one want to use Abel's test, is $$ \sum_{n=1}^\infty |\sin n|^n\tag{2}$$ convergent？ Thank you very much
2016
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66
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22 answers

Nuking the Mosquito — ridiculously complicated ways to achieve very simple results

Here is a toned down example of what I'm looking for: Integration by solving for the unknown integral of $f(x)=x$: $$\int x \, dx=x^2-\int x \, dx$$ $$2\int x \, dx=x^2$$ $$\int x \, dx=\frac{x^2}{2}$$ Can anyone think of any more examples? P.S.…
Aidan F. Pierce
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66
votes
6 answers

Proving that $m+n\sqrt{2}$ is dense in $\mathbb R$

I am having trouble proving the statement: Let $$S = \{m + n\sqrt 2 : m, n \in\mathbb Z\}$$ Prove that for every $\epsilon > 0$, the intersection of $S$ and $(0, \epsilon)$ is nonempty.
user11135
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66
votes
12 answers

Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $

I know that the harmonic series $$\sum_{k=1}^{\infty}\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots + \frac{1}{n} + \cdots \tag{I}$$ diverges, but what about the alternating harmonic series…
Isaac
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66
votes
3 answers

Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$?

I have a couple of questions about tensor products: Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$? Why is an element of $V^{*\otimes m}\otimes V^{\otimes n}$ the same thing as a multilinear map $V^m \to V^{\otimes n}$? What is the…
Eric Auld
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66
votes
4 answers

Mathematicians don't quit, they fade away

Edit: This question is now closed for being not related to math, but many people pointed out that becoming an actuary is one of the most viable career path for someone with skills in pure math. Noone I've ever talked to knows what mathematicians do…
Brian Rushton
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66
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5 answers

Showing that $\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$, when $f$ is even

I have a question: Suppose $f$ is continuous and even on $[-a,a]$, $a>0$ then prove that $$\int\limits_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm dx = \int\limits_0^a f(x) \mathrm dx$$ How can I do this? Don't know how to start.
James
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