## Top new questions this week:

### Examples of mathematical results discovered "late"

What are examples of mathematical results that were discovered surprisingly late in history? Maybe the result is a straightforward corollary of an established theorem, or maybe it's just so simple …

(soft-question) (math-history) (big-list)

### Do we need to formally teach the Greek Alphabet?

This is a question that I am purely interested in because I think we never thought about this before in Mathematics education... or even so was not discussed. When did we learn the Greek alphabets …

(soft-question) (teaching)

### What is exponentiation?

Is there an intuitive definition of exponentiation? In elementary school, we learned that $$a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times})$$ where $b$ is an integer. Then later on …

(algebra-precalculus) (soft-question) (exponentiation)

### What exactly are fractals

I have always been amazed by things like the Mandelbrot set. I share the view of most that it and the Koch snowflake are absolutely beautiful. I decided to get a deeper more mathematical knowledge of …

(fractals)

### An exotic sequence

Let $a=\frac{1+i\sqrt 7}{2}$ and $u_n=\Re(a^n)$ show that $(|u_n|)\to +\infty$ I tried to calculate $\Re(a^n)$ but it's seems to be very difficult to do that. I think basics method does not works …

(calculus) (real-analysis) (sequences-and-series)

I was challenged to prove this identity $$\int_0^\infty\frac{\log\left(\frac{1+x^{4+\sqrt{15\vphantom{\large A}}}}{1+x^{2+\sqrt{3\vphantom{\large A}}}}\right)}{\left(1+x^2\right)\log … (calculus) (integration) (definite-integrals) (logarithms) (improper-integrals)  asked by Zakharia Stanley 17 votes  answered by Vladimir Reshetnikov 14 votes ### Geometric explanation of \sqrt 2 + \sqrt 3 \approx \pi Just curious, is there a geometry picture explanation to show that \sqrt 2 + \sqrt 3  is close to  \pi ? (geometry) (trigonometry) (approximation)  asked by ahala 15 votes  answered by Shuchang 12 votes ## Greatest hits from previous weeks: ### Integral \int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \ \mathrm dx I need help with this integral:$$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx. The integrand graph looks like this: $\hspace{1in}$ The …

(calculus) (integration) (definite-integrals) (contour-integration) (closed-form)

### Is there another simpler method to solve this elementary school math problem?

I am teaching an elementary student. He has a homework as follows. There are 16 students who use either bicycles or tricycles. The total number of wheels is 38. Find the number of students using …

(homework) (linear-algebra) (arithmetic)

### Putnam 2008 A4 without the integral test

Problem $\big[\text{Putnam}\;2008\;\mathrm{A}4\big]:$ Define $f : \mathbb{N}\to\mathbb{R}$ such that $f(n)= n \log n \log_2 n \cdots \log_{\kappa(n)} n$ where $\kappa (n)$ is the largest …

(sequences-and-series) (convergence) (contest-math)

### Group Structure on $\Bbb R$

$(\Bbb R,+)$ is a topological group. Is there any other group structure on $\Bbb R$ such that it is still a topological group and this group is not isomorphic to $(\Bbb R,+)$ ? Refer to …

(real-analysis) (general-topology) (topological-groups) (real-numbers)
### Does the sequence $\sin(n!\pi^2)$ converge or diverge?
Does the sequence $\sin(n!\pi^2)$ converge or diverge?