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1500 questions
145
votes
16 answers
Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$
Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$
where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number. The $A(p,q)$'s are known as alternating Euler sums.
Can someone provide a nice proof that
$$A(1,1) =…
Mike Spivey
- 56,818
145
votes
0 answers
If polynomials are almost surjective over a field, is the field algebraically closed?
Let $K$ be a field. Say that polynomials are almost surjective over $K$ if for any nonconstant polynomial $f(x)\in K[x]$, the image of the map $f:K\to K$ contains all but finitely many points of $K$. That is, for all but finitely many $a\in K$,…
Eric Wofsey
- 342,377
145
votes
9 answers
Normal subgroup of prime index
Generalizing the case $p=2$ we would like to know if the statement below is true.
Let $p$ the smallest prime dividing the order of $G$. If $H$ is a subgroup of $G$ with index $p$ then $H$ is normal.
Sigur
- 6,620
144
votes
18 answers
How to put 9 pigs into 4 pens so that there are an odd number of pigs in each pen?
So I'm tutoring at the library and an elementary or pre K student shows me a sheet with one problem on it:
Put 9 pigs into 4 pens so that there are an odd number of pigs in each pen.
I tried to solve it and failed! Does anybody know how to solve…
zerosofthezeta
- 5,710
144
votes
1 answer
Identification of a curious function
During computation of some Shapley values (details below), I encountered the following function:
$$
f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}},
$$
where $p_0 > 0$ and $p_{k+1} > p_k$ for all $k$. In…
Yuval Filmus
- 57,953
144
votes
6 answers
Studying for the Putnam Exam
This is a question about studying for the Putnam examination (and, secondarily, other high-difficulty proof-based math competitions like the IMO). It is not about the history of the competition, the advisability of participating, the career…
Potato
- 41,411
144
votes
10 answers
Is the blue area greater than the red area?
Problem:
A vertex of one square is pegged to the centre of an identical square, and the overlapping area is blue. One of the squares is then rotated about the vertex and the resulting overlap is red.
Which area is greater?
Let the area of each…
Mr Pie
- 9,726
144
votes
6 answers
Why is integration so much harder than differentiation?
If a function is a combination of other functions whose derivatives are known via composition, addition, etc., the derivative can be calculated using the chain rule and the like. But even the product of integrals can't be expressed in general in…
Venge
- 1,621
143
votes
7 answers
Construction of a Borel set with positive but not full measure in each interval
I was wondering how one can construct a Borel set that doesn't have full measure on any interval of the real line but does have positive measure everywhere.
To be precise, if $\mu$ denotes Lebesgue measure, how would one construct a Borel set $A…
user1736
- 8,993
143
votes
22 answers
List of interesting math podcasts?
mathfactor is one I listen to. Does anyone else have a recommendation?
Tim
- 119
143
votes
0 answers
Mondrian Art Problem Upper Bound for defect
Divide a square of side $n$ into any number of non-congruent rectangles. If all the sides are integers, what is the smallest possible difference in area between the largest and smallest rectangles?
This is known as the Mondrian Art Problem. For…
Ed Pegg
- 21,868
143
votes
9 answers
What exactly is the difference between a derivative and a total derivative?
I am not too grounded in differentiation but today, I was posed with a supposedly easy question $w = f(x,y) = x^2 + y^2$ where $x = r\sin\theta $ and $y = r\cos\theta$ requiring the solution to $\partial w / \partial r$ and $\partial w / \partial…
Chibueze Opata
- 1,565
142
votes
15 answers
Why is $1$ not a prime number?
Why is $1$ not considered a prime number?
Or, why is the definition of prime numbers given for integers greater than $1$?
bryn
- 10,045
141
votes
13 answers
Can someone explain Gödel's incompleteness theorems in layman terms?
Can you explain it in a fathomable way at high school level?
Hyperbola
- 2,485
141
votes
11 answers
Why is $\infty \cdot 0$ not clearly equal to $0$?
I did a bit of math at school and it seems like an easy one - what am I missing?
$$n\times m = \underbrace{n+n+\cdots +n}_{m\text{ times}}$$
$$\quad n\times 0 = \underbrace{0 + 0 + \cdots+ 0}_{n\text{ times}} = 0$$
(i.e add $0$ to $0$ as many times…
Ashley Schroder
- 1,537