# Mathematical function that converges towards $7$?

My friends and I are finishing High School in Denmark. We have to do a math poster for some school activity, where the poster needs to have something to do with the number $7$. So my question is: does someone know a cool mathematical function that converges towards $7$? We covered Calculus III, so we should be able to understand a little math!

• $\frac{42}{\pi^2} \sum_{n=1}^\infty \frac{1}{n^2}$ – Gregory Apr 25 '17 at 20:02
• en.wikipedia.org/wiki/Fano_plane ... $7$ points & $7$ lines (each line containing $3$ points) ... a fundemental object of finite projective geometry. – Donald Splutterwit Apr 25 '17 at 20:07
• A function does not converge. – Carsten S Apr 26 '17 at 8:57

One with the Fibonacci sequence:

$$\lim_{n\to\infty}\frac{4F_{n+1}^2 - 4F_{n+1}F_n + 3F_n^2}{F_n^2} = \lim_{n\to\infty}\left(2\frac{F_{n+1}}{F_n} - 1\right)^2 + 2 = 7.$$

If integrals are acceptable, this one is nice:

$$7= \frac{1}{\displaystyle\int_0^1 \frac{x^4(1-x)^4}{1+x^2} \, dx +\pi-3}$$

(source: Wikipedia)

• Whoa! The famous "$\frac {22} {7}- \pi$" integral. Clever ! (+unity) – user399078 Apr 26 '17 at 9:02

Here is one that looks crazy at first, but it is actually quite simple:

$$\lim\limits_{N\to\infty}\left[\frac N{3\pi}\sin\left(\frac{42}{N}\sum\limits_{k=1}^N\left(\frac{\pi}{\pi+2}\right)^k\right)\right] = 7.$$

What about $7+a_n$ where $a_n$ converges to $0$? You could then proceed to find any fancy function you want, like

$$a_n=7+\frac{7}{7^{7n+7}}$$

converges to $7$.

Or, given a sequence $b_n$ converging to $L\neq 0$, the sequence $\frac7Lb_n$ converges to $7$, like

$$a_n=\left(7-\frac{7}{7}\right)\sum_{k=0}^n\frac{1}{7^k}$$

converges to $7$.

Some other cool facts about $7$ can be found on Wikipedia's page on 7.

$$8-\frac 87+\frac{8}{49}-\frac{8}{343}+...$$

Or, maybe

$$\sqrt[n]{\sum_{r=0}^n\binom{n}{r}6^r}$$

$111_2 = 7$

$0.111\ldots_8 = 1/7$.

• And to answer the question with this, the function would be $f(n) = 1 / {0.\underbrace{111\ldots1_8}_{n}}$. – 6005 Apr 26 '17 at 0:30

There is Newton's method for square root. Let $$x_{n+1}=\frac12\left(x_n+\frac{49}{x_n}\right)$$ for any $x_0 > 0$. Then $x_n \to 7$.

$$\lim_{x \to 0} \frac{\sin(21x)}{3x}=7$$

Not a strict answer to the question, but still related to calculs : the Borwein integrals. Let us denote by $\operatorname{sinc}$ the sinus cardinal function $\operatorname{sinc}x=\frac{\sin x}{x}$ and consider the following integrals defined for any natural number $n$: $$B_n=\int_0^\infty \left(\prod_{k=0}^n \operatorname{sinc}\frac{x}{2k+1}\right)\,\mathrm{d}x.$$ Then you have $B_n=\frac{\pi}{2}$ if and only if $n<7$.

Simple:

$$\lim_{x \rightarrow\infty} \frac{n}{x}+7=7$$ where $n =$ any number