Leibniz’s Formula for π

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.

Credits: gizmodo.com

Credits: gizmodo.com

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.

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