Mathematical function that converges towards $7$?
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)
Here is one that looks crazy at first, but it is actually quite simple:
$$\lim\limits_{N\to\infty}\left[\frac N{3\pi}\sin\left(\frac{42}{N}\sum\limits_{k=1}^N\left(\frac{\pi}{\pi+2}\right)^k\right)\right] = 7. $$