Mathematical function that converges towards $7$?

Question

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!

Answer 1

One with the Fibonacci sequence:

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

Answer 2

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)

Answer 3

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

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

Answer 4

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 $7$.

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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 $7$.

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

Answer 5

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

Or, maybe

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

Answer 6

$111_2 = 7$

$0.111\ldots_8 = 1/7$.

Answer 7

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 $x_n \to 7$.

Answer 8

Simple:

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

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Answer 9

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 $n<7$.

Answer 10

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