Hot answers tagged constants
14
The number $e$ is the number such that the area enclosed by the region bounded by $x=1$ on the left-hand side, the $X$ axis from below, $y=1/x$ from above and $x=e$ on the right-hand side is $1$. Hence, $e$ can be defined as the length (you need to add $1$, since you measure the length from $1$) you need to move along $X$ axis such that the area of the above ...
8
A different take on the 'classical' limit that I think is my favorite way of thinking about $e$ recreationally (and a remarkably useful approximation for many games): "I take a six-sided die and roll it six times. What are the odds I never roll '1' in those six rolls? Okay, now I take a twenty-sided die and roll it twenty times. What are the odds I never ...
7
Decompose the product on the right as
$$\prod_{\text{primes}\; p\\ \text{ of the form }4k+1}\left(1+\frac{1}{p}\right)\prod_{\text{primes}\; p\\ \text{ of the form }4k+3}\left(1-\frac{1}{p}\right)$$
Consider an odd integer $n=2m+1$. It is "easy to see" that if primes of the form $4k+3$ appear in its prime number decomposition an even number of times, ...
6
Our goal is to evaluate the sum $$\sum_{k=0}^{\infty}\left(\frac{2^{4k+1}+1}{\left(8k+1\right)}+\frac{2^{4k+2}-1}{2^{2}\left(8k+3\right)}-\frac{2^{4k+3}-1}{2^{4}\left(8k+5\right)}-\frac{2^{4k+4}+1}{2^{6}\left(8k+7\right)}\right)2^{-8k}.$$ We split this into two different convergent sums, ...
5
The notion of period, which is introduced by Kontsevich and Zagier, would partially give a negative answer to your question. According to this article, it is now known whether $e$ is a period or not, though it is conjecturedly not a peroid. In particular, $e$ seems not to arise as an area or a length of a geometric figure defined by an algebraic equation.
...
5
Consider the combination
\begin{align}f_N=\frac{\Gamma\left(N+\frac{2}{5}\right)\Gamma\left(N+\frac{3}{5}\right)}{\Gamma\left(N+\frac{1}{5}\right)\Gamma\left(N+\frac{4}{5}\right)}=
\frac{\left(5N-2\right)\cdot\left(5N-3\right)}{\left(5N-1\right)\cdot\left(5N-4\right)}f_{N-1}=\ldots=\\=
...
3
$$e\approx 2.71828182846$$
Consider the equation:
$$f(n)=(1+\frac{1}{n})^n$$
As $n$ gets larger and larger, notice what the result approaches.
$$f(1)=2$$
$$f(2)=2.25$$
$$f(3)\approx2.3703703$$
$$...$$
$$f(100)\approx2.7048138$$
$$...$$
2
This one is not visual or graphical, but may be easiest to understand for a non-mathematician.
Suppose you have $\$1000$ and you want to put it in a bank account. You have picked a bank that, besides giving you an absurd interest over your money, gives you a choice between several interest schemes:
An annual interest of $100\%$.
An interest of $50\%$, but ...
2
See the update for the answer to question. Below follows a proof that the series are the same.
Here is how I did it:
Note that every positive integer can be written either as an even or odd number (of the form $2k$ for $k \in \mathbb{N}$ and $2k+1$ respectively). Then, since your sum goes through all the integers, it is the same as:
$$
\begin{align}
...
1
I'm surprised the terminology is different in French, but Wikipédia seems to agree:
Un nombre positif est un nombre qui est supérieur (au sens de : supérieur ou égal)
à zéro [...] Zéro est un nombre réel positif [...]
Lorsqu'un nombre est positif et non nul, il est dit strictement positif.
The most common usage in English is that zero is neither ...
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