Mathematics Weekly Newsletter
Mathematics Weekly Newsletter

Top new questions this week:

Why can you chose how to align infinitely long equations when adding them?

I saw in a video this proof: Take this equation: $$f=1+\frac12+\frac14+\cdots$$ and do this: $$\begin{align} f&=1+1/2+1/4+\cdots\\ -\quad f/2&=\quad\:\:\:1/2+1/4+\cdots\\ ...

(sequences-and-series) (convergence) (infinity)  
asked by QxQ 33 votes
answered by Johanna 34 votes

Inequality from Chapter 5 of the book *How to Think Like a Mathematician*

This is from the book How to think like a Mathematician, How can I prove the inequality $$\sqrt[\large 7]{7!} < \sqrt[\large 8]{8!}$$ without complicated calculus? I tried and finally obtained ...

(algebra-precalculus) (inequality)  
asked by Gwydyon 30 votes
answered by Crostul 77 votes

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. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ ...

(number-theory) (exponentiation) (perfect-powers)  
asked by mathlove 19 votes

Why is Gödel's Second Incompleteness Theorem important?

Given that the consistency of a system can be proven outside of the given formal system, Gödel says, It must be noted that proposition XI... represents no contradiction to the formalities ...

(logic) (formal-languages) (proof-theory) (incompleteness)  
asked by MadScientist 15 votes
answered by Mauro ALLEGRANZA 9 votes

Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc... & Representation Theory of Special Functions

Many special functions of mathematical physics can be understood from the point of view of the representation theory of lie groups. An example of the power of this viewpoint is given in my question ...

(differential-equations) (representation-theory) (special-functions) (lie-groups)  
asked by bolbteppa 14 votes

Abscissa, Ordinate and ?? for z-axis?

Like x-axis is abscissa, y-axis is ordinate what is z-axis called? It is one of basic doubts from my childhood.

asked by Pervez Alam 14 votes
answered by Murp 15 votes

Regular way to fill a $1\times1$ square with $\frac{1}{n}\times\frac{1}{n+1}$ rectangles?

The series $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}=1$$ suggests it might be possible to tile a $1\times1$ square with nonrepeated rectangles of the form $\frac{1}{n}\times\frac{1}{n+1}$. Is there a ...

(sequences-and-series) (visualization) (egyptian-fractions)  
asked by alex.jordan 14 votes

Greatest hits from previous weeks:

What is the integral of 0?

I am trying to convince my friend that the integral of 0 is C, where C is an arbitrary constant. He can't seem to grasp this concept. Can you guys help me out here? He keeps saying it is 0.

asked by HowardRoark 12 votes
answered by Sam DeHority 16 votes

Can a coin with an unknown bias be treated as fair?

This morning, I wanted to flip a coin to make a decision but no coins were in reach. There was however an SD card on my desk: Given that I don't know the bias of this SD card, would flipping it be ...

(probability) (random) (philosophy)  
asked by Andrew Cheong 135 votes
answered by josinalvo 59 votes

Can you answer these?

Elliptic and Fredholm partial differential operators

As I learn from the comments to this question a non elliptic operator on a compact manifold can not be a fredholm operator. On the other hand it seems that the following operator is fredholm(At ...

(pde) (dynamical-systems) (differential-operators) (elliptic-equations)  
asked by Ali Taghavi 5 votes

Can anyone explain what is the intuition behind the following definition of $p \Vdash^* \phi $?

Can anyone explain what is the intuition behind the following definition? I know that the sign $p \Vdash \phi(x_1,...,x_n)$ somehow suppose to tell me that for any generic filter which contains ...

(logic) (set-theory) (forcing)  
asked by user135172 6 votes

What about $\mathrm{Spec}(\mathbf{Q})$?

I've heard a lot about $\mathrm{Spec}(\mathbf{Q})$ (see for example Minhyong Kim's answer here), but $\mathbf{Q}$ is a field. So isn't $\mathrm{Spec}(\mathbf{Q})$ trivial? What's the point of studying ...

(commutative-algebra) (prime-ideals)  
asked by NoClueWhatI'mDoingHere 4 votes
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