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

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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KS Teng

KS has been teaching H2/H1 Mathematics and IB mathematics for the more than 10 years. Having taught students from all various Junior Colleges, KS adapts to students' abilities and help them better understand the topics. As someone who loves to teach mathematics and sees it as a truly useful tool in life, KS seeks to enable students to appreciate math. Therefore, his tuition mission is to motivate and cultivate students to be independent and confident thinkers.

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Leibniz’s Formula for π

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