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.

Leave a Comment

3 × four =

Contact Us

CONTACT US We would love to hear from you. Contact us, or simply hit our personal page for more contact information

Not readable? Change text. captcha txt

Start typing and press Enter to search