Here is some sharing over a very fascinating constant $\pi$. We know this is $\frac {22}{7}$ (up to 3 sig. fig. accuracy) or even $\frac {355}{113}$ (up to 7 sig. fig. accuracy). One of my favourite way of deriving this never-ending amazing constant is to use the Leibniz Formula.

A-level students should be able to solve it on your own. We first find the series expansion for $\frac {1}{1+t^2}$ with our MF15 $(1+x)^n$ formula.

$\frac {1}{1+t^2}=1-t^{2}+t^{4}-t^{6}+t^{8}+...$

After this, note that $\int \frac {1}{1+t^2} dt = tan^{-1}t$.

We can then integrate the series to find

$tan^{-1}t=t- \frac {t^{3}}{3} + \frac {x^{5}}{5}- \frac {t^{7}}{7} + \frac {x^{9}}{9}...$

Finally, we just need to substitute $t=1$ and you should find $\pi$ somewhere.