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1500 questions
67
votes
14 answers
Express 99 2/3% as a fraction? No calculator
My 9-year-old daughter is stuck on this question and normally I can help her, but I am also stuck on this! I have looked everywhere to find out how to do this but to no avail so any help/guidance is appreciated:
The possible answers are:
$1…
lara400
- 886
- 1
- 7
- 9
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
- 151,597
67
votes
3 answers
Closed form solution for $\sum_{n=1}^\infty\frac{1}{1+\frac{n^2}{1+\frac{1}{\stackrel{\ddots}{1+\frac{1}{1+n^2}}}}}$.
Let
$$
\text{S}_k = \sum_{n=1}^\infty\cfrac{1}{1+\cfrac{n^2}{1+\cfrac{1}{\ddots1+\cfrac{1}{1+n^2}}}},\quad\text{$k$ rows in the continued fraction}
$$
So for example, the terms of the sum $\text{S}_6$…
Simon Parker
- 4,421
66
votes
3 answers
What use is the Yoneda lemma?
Although I know very little category theory, I really do find it a pretty branch of mathematics and consider it quite useful, especially when it comes to laying down definitions and unifying diverse concepts. Many of the tools of category theory…
Alex Becker
- 61,883
66
votes
7 answers
Evaluating the indefinite integral $ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $
I have been having extreme difficulties with this integral. I would appreciate any and all help.
$$
\int \sqrt{\tan x} ~ \mathrm{d}{x}.
$$
A is for Ambition
- 2,138
66
votes
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
- 1,073
66
votes
3 answers
Expectation of Minimum of $n$ i.i.d. uniform random variables.
$X_1, X_2, \ldots, X_n$ are $n$ i.i.d. uniform random variables. Let $Y = \min(X_1, X_2,\ldots, X_n)$. Then, what's the expectation of $Y$(i.e., $E(Y)$)?
I have conducted some simulations by Matlab, and the results show that $E(Y)$ may equal to…
jet
- 763
66
votes
4 answers
Open maps which are not continuous
What is an example of an open map $(0,1) \to \mathbb{R}$ which is not continuous? Is it even possible for one to exist? What about in higher dimensions? The simplest example I've been able to think of is the map $e^{1/z}$ from $\mathbb{C}$ to…
math student
- 661
66
votes
4 answers
Areas versus volumes of revolution: why does the area require approximation by a cone?
Suppose we rotate the graph of $y = f(x)$ about the $x$-axis from $a$ to $b$. Then (using the disk method) the volume is $$\int_a^b \pi f(x)^2 dx$$ since we approximate a little piece as a cylinder. However, if we want to find the surface area,…
Eric O. Korman
- 19,927
66
votes
6 answers
Does $ \int_0^{\infty}\frac{\sin x}{x}dx $ have an improper Riemann integral or a Lebesgue integral?
In this wikipedia article for improper integrals,
$$
\int_0^{\infty}\frac{\sin x}{x}dx
$$
is given as an example for the integrals that have an improper Riemann integral but do not have a (proper) Lebesgue integral. Here are my questions:
Why does…
user9464
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
- 13,375
66
votes
7 answers
What is the antiderivative of $e^{-x^2}$
I was wondering what the antiderivative of $e^{-x^2}$ was, and when I wolfram alpha'd it I got $$\displaystyle \int e^{-x^2} \textrm{d}x = \dfrac{1}{2} \sqrt{\pi} \space \text{erf} (x) + C$$
So, I of course didn't know what this $\text{erf}$ was…
Phaptitude
- 2,297
66
votes
8 answers
Example of Partial Order that's not a Total Order and why?
I'm looking for a simple example of a partial order which is not a total order so that I can grasp the concept and the difference between the two.
An explanation of why the example is a partial order but not a total order would also be greatly…
eZanmoto
- 779
66
votes
5 answers
Is the box topology good for anything?
In point-set topology, one always learns about the box topology: the topology on an infinite product $X = \prod_{i \in I} X_i$ generated by sets of the form $U = \prod_{i \in I} U_i$, where $U_i \subset X_i$ is open. This seems naively like a…
Nate Eldredge
- 101,664
66
votes
5 answers
When is the closure of an intersection equal to the intersection of closures?
We know $\overline{\bigcap A_{\alpha}}\subseteq\bigcap\overline{A}_{\alpha} $, but when is the reverse inclusion true? Can you give some properties of the underlying space that would guarantee this?
Forever Mozart
- 9,534