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1500 questions
90
votes
15 answers
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 being repurposed in an effort to cut down on duplicates, see…
Matt Nashra
- 1,245
90
votes
9 answers
Why do mathematicians sometimes assume famous conjectures in their research?
I will use a specific example, but I mean in general. I went to a number theory conference and I saw one thing that surprised me: Nearly half the talks began with "Assuming the generalized Riemann Hypothesis..." Almost always, the crux of their…
Joseph DiNatale
- 2,887
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- 38
90
votes
3 answers
Conjecture $\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi$
$$\int_0^1\frac{\mathrm dx}{\sqrt{1-x}\ \sqrt[4]x\ \sqrt[4]{2-x\,\sqrt3}}\stackrel?=\frac{2\,\sqrt2}{3\,\sqrt[8]3}\pi\tag1$$ The equality numerically holds up to at least $10^4$ decimal digits. Can we prove that the equality is exact? An equivalent…
Vladimir Reshetnikov
- 32,880
90
votes
2 answers
Conjecture $\int_0^1\frac{dx}{\sqrt[3]x\,\sqrt[6]{1-x}\,\sqrt{1-x\left(\sqrt{6}\sqrt{12+7\sqrt3}-3\sqrt3-6\right)^2}}=\frac\pi9(3+\sqrt2\sqrt[4]{27})$
Let $$\alpha=\sqrt{6}\ \sqrt{12+7\,\sqrt3}-3\,\sqrt3-6\,=\,\big(2+\sqrt{3}\big) \big(\sqrt{2} \sqrt[4]{27}-3\big)\,=\,\frac{3\sqrt{3}}{3+\sqrt2\ \sqrt[4]{27}}.\tag1$$ Note that $\alpha$ is the unique positive root of the polynomial equation…
Vladimir Reshetnikov
- 32,880
90
votes
3 answers
Under what conditions the quotient space of a manifold is a manifold?
There are many operations we can do with topological spaces that when we apply to topological manifolds gives us back topological manifolds. The disjoint union and the product are examples of that. Another common operation is to take the quotient by…
Gold
- 28,441
90
votes
9 answers
Generalized Euler sum $\sum_{n=1}^\infty \frac{H_n}{n^q}$
I found the following formula $$\sum_{n=1}^\infty \frac{H_n}{n^q}= \left(1+\frac{q}{2} \right)\zeta(q+1)-\frac{1}{2}\sum_{k=1}^{q-2}\zeta(k+1)\zeta(q-k)$$ and it is cited that Euler proved the formula above , but how ? Do there exist other proofs ?…
Zaid Alyafeai
- 14,883
90
votes
6 answers
Why is the determinant the volume of a parallelepiped in any dimensions?
For $n = 2$ , I can visualize that the determinant $n \times n$ matrix is the area of the parallelograms by actually calculating the area by coordinates. But how can one easily realize that it is true for any dimensions?
ahala
- 3,170
90
votes
1 answer
Conjectured formula for the Fabius function
The Fabius function is the unique function ${\bf F}:\mathbb R\to[-1, 1]$ satisfying the following conditions: a functional–integral equation $\require{action} \require{enclose}{^{\texttip{\dagger}{a poet or philosopher could say "it knows and…
Vladimir Reshetnikov
- 32,880
90
votes
10 answers
Is mathematics just a bunch of nested empty sets?
When in high school I used to see mathematical objects as ideal objects whose existence is independent of us. But when I learned set theory, I discovered that all mathematical objects I was studying were sets, for example: $ 0 = \emptyset $ What…
Vinicius L. Deloi
- 1,275
90
votes
7 answers
Prove that $i^i$ is a real number
According to WolframAlpha, $i^i=e^{-\pi/2}$ but I don't know how I can prove it.
Isaac
- 1,139
90
votes
5 answers
How to solve these two simultaneous "divisibilities" : $n+1\mid m^2+1$ and $m+1\mid n^2+1$
Is it possible to find all integers $m>0$ and $n>0$ such that $n+1\mid m^2+1$ and $m+1\,|\,n^2+1$ ? I succeed to prove there is an infinite number of solutions, but I cannot progress anymore. Thanks !
user14479
90
votes
10 answers
Would you ever stop rolling the die?
You have a six-sided die. You keep a cumulative total of your dice rolls. (E.g. if you roll a 3, then a 5, then a 2, your cumulative total is 10.) If your cumulative total is ever equal to a perfect square, then you lose, and you go home with…
Newb
- 18,033
90
votes
8 answers
Is there a size of rectangle that retains its ratio when it's folded in half?
A hypothetical (and maybe practical) question has been nagging at me. If you had a piece of paper with dimensions 4 and 3 (4:3), folding it in half along the long side ( once ) would result in 2 inches and 3 inches (2:3), which wouldn't retain its…
Pyraminx
- 1,011
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- 10
89
votes
3 answers
Ramanujan log-trigonometric integrals
I discovered the following conjectured identity numerically while studying a family of related integrals. Let's set $$ R^{+}:= \frac{2}{\pi}\int\limits_{0}^{\pi/2}\sqrt[\normalsize{8}]{x^2 + \ln^2\!\cos x} \sqrt{ \frac{1}{2}+\frac{1}{2}\sqrt{…
Olivier Oloa
- 123,429
89
votes
31 answers
What are some conceptualizations that work in mathematics but are not strictly true?
I'm having an argument with someone who thinks it's never justified to teach something that's not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is to iteratively learn and unlearn along the way. I'm…
MGA
- 9,869