Mathematics Weekly Newsletter
Mathematics Weekly Newsletter

Top new questions this week:

Why does $ \int_0^1 \lceil { x\sin({1 \over x})} \rceil = 1 - \frac{\log(4)}{2\pi} $?

One time I was bored and played around a bit with integrals and wolfram alpha and tested the following integral: http://www.wolframalpha.com/input/?i=integral_0%5E1+ceil%28x*sin%281%2Fx%29%29 Note: ...

(calculus) (integration)  
asked by Imago 24 votes
answered by A.G. 37 votes

Is $1992! - 1$ prime?

Consider the factorials, defined inductively by $1! = 0! = 1$ and $n! = n\cdot(n-1)!$ for $n \geq 2$. Question: Is $1992!-1$ a prime number? The question is from a book, maybe is contest math ...

(number-theory) (elementary-number-theory) (prime-numbers)  
asked by Thailandasw 23 votes
answered by Julián Aguirre 16 votes

A circle in the plane contains at most four lattice points?

Let $\cal C$ be a circle in ${\mathbb R}^2$ : $\cal C=\lbrace (x,y)\in{\mathbb R}^2 | (x-x_0)^2+(y-y_0)^2=r^2\rbrace$ for some constants $x_0,y_0,r$. What is the maximal number of points that can ...

(euclidean-geometry)  
asked by Ewan Delanoy 20 votes
answered by Mark Bennet 50 votes

Evaluate $\displaystyle\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^\frac{1}{n} \ \mathrm{d}x$

Evaluate $$\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^\frac{1}{n} \ \mathrm{d}x$$ I simplified the limit to $$\dfrac{1}{2}\lim_{n \to \infty} \int_{0}^{\frac{1}{2}} ...

(calculus) (integration) (limits) (definite-integrals)  
asked by Samurai 19 votes
answered by user2566092 9 votes

About the integral $\int_{-1}^1 \frac{1}{\pi^2+(2 \operatorname{arctanh}(x))^2} \, dx=\frac{1}{6} $

Here is a question that naturally arose in the study of some specific integrals. I'm curious if for such integrals are known nice real analysis tools for calculating them (including here all possible ...

(calculus) (real-analysis) (integration) (reference-request) (definite-integrals)  
asked by Chris's sis the artist 18 votes
answered by M.N.C.E. 5 votes

show this sequence inequality $x_{2^n}$

Define the sequence $\{x_{n}\}$ recursively by $x_{1}=1$ and $$\begin{cases} x_{2k+1}=x_{2k}\\ x_{2k}=x_{2k-1}+x_{k} \end{cases}$$ Prove that $$x_{2^n}>2^{\frac{n^2}{4}}$$ I have ...

(sequences-and-series)  
asked by Thailandasw 16 votes
answered by Eric Naslund 7 votes

How to divide by a matrix

I found a question in an old exam, where the function $\phi(z) := \frac{\exp(z) - 1}{z}$ is given. Now we evaluate $\phi(\mathbf{A})$. But how do I divide by a matrix? I already thought about ...

(linear-algebra) (matrices) (functions)  
asked by Steven 15 votes
answered by uranix 25 votes

Greatest hits from previous weeks:

What is the importance of eigenvalues/eigenvectors?

What is the importance of eigenvalues/eigenvectors?

(linear-algebra) (soft-question) (eigenvalues-eigenvectors)  
asked by Ryan 82 votes
answered by Arturo Magidin 114 votes

Pedagogy: How to cure students of the "law of universal linearity"?

One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”: $$ \frac{1}{a+b} ...

(algebra-precalculus) (education) (teaching)  
asked by Peter LeFanu Lumsdaine 249 votes
answered by goblin 122 votes

Can you answer these?

limit of a region of integration in $\mathbb{R}^2$ approaches a line

I am trying to follow the derivation of derivatives in a paper published in some japanese journal but there seems to be a mistake in the proof. I will present the problem in 2D and in 2 variables so ...

(calculus) (integration) (derivatives)  
asked by Tomas Jorovic 6 votes

Curtains and groups

This picture is a copy of the pattern on my curtains. The points of a hexagonal lattice are each coloured with one of four possible colours. It has translational symmetry in two directions: a ...

(group-theory)  
asked by octopus 13 votes

A difficult logarithmic integral ${\Large\int}_0^1\log(x)\,\log(2+x)\,\log(1+x)\,\log\left(1+x^{-1}\right)dx$

A friend of mine shared this problem with me. As he was told, this integral can be evaluated in a closed form (the result may involve polylogarithms). Despite all our efforts, so far we have not ...

(integration) (definite-integrals) (logarithms) (closed-form) (polylogarithm)  
asked by Laila Podlesny 11 votes
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