In order to cleanse this board of brainlets and pop sci faggots, everyone post your favorite non-trivial formula for pi

All 10/10 answers

Pi == ((2^n/n) Integrate[Sin[t]^n/t^n, {t, 0, Infinity}])/ Sum[((-1)^k (n - 2 k)^(n - 1))/(k! (n - k)!), {k, 0, Floor[(n - 1)/2]}] /; Element[n, Integers] && n > 0

The formulas are derived using calculus and trigonometry. It's not a case of evaluating the formula and checking the digits against pi

fug :DD my formula didn't work

[eqn]1 + 1/4 + ... = pi^2/6[/eqn]

Forgot a \ there

yes

learn [math]\LaTeX[/math] for fucks sake

>What's the proof that each of these actually amount to Pi? Who's to say that every digit is identical and spot on except for the 500^2'th digit?

This is done using calculus, or if you want to be fancy then you can call it real anal.

When you do anal for real, you define a sequence and you "think of" a number that it converges to. Then, through some weird mathematical machinery, you can prove that your given sequence does converge to that number.

In this case, these people constructed this sequences probably by using some method they found for approximating Pi, and then naturally you think that if you were to continue this approximation you would, at infinity, reach Pi.

Then, after you think that, you find the definition of convergence and prove rigorously that indeed that is the case by doing a lot of gay inequalities.

Someone found a formula similar to OP's which yields the first billion digits but then begins to differ from pi. The trick is to start with a well known identity such as the gaussian integral [math]\left(\int e^{-x^2}\right)^2 = 2\pi[/math] and approximate it with a discrete series, e.g. writing the integral as a Riemann sum.

[math]\pi = \text{ }^{\color{#571da2}{\displaystyle\text{W}}}\text{ }^{^{^{^{\color{#462eb9}{\displaystyle\text{h}}}}}}\text{ }^{^{^{^{^{^{^{\color{#3f47c8}{\displaystyle\text{y}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{\color{#3f62cf}{\displaystyle\text{ }}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#437ccc}{\displaystyle\text{i}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#4b90bf}{\displaystyle\text{s}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#56a0ae}{\displaystyle\text{ }}}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#62ab99}{\displaystyle\text{t}}}}}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#71b484}{\displaystyle\text{h}}}}}}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#82ba70}{\displaystyle\text{i}}}}}}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#96bc5f}{\displaystyle\text{s}}}}}}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#a9bd52}{\displaystyle\text{ }}}}}}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#bcbb48}{\displaystyle\text{o}}}}}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#ceb541}{\displaystyle\text{n}}}}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#dcab3c}{\displaystyle\text{ }}}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{^{^{^{\color{#e39938}{\displaystyle\text{/}}}}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{^{^{^{\color{#e68033}{\displaystyle\text{s}}}}}}}}}}}}\text{ }^{^{^{^{^{^{^{\color{#e3632d}{\displaystyle\text{c}}}}}}}}}\text{ }^{^{^{^{\color{#de4227}{\displaystyle\text{i}}}}}}\text{ }^{\color{#da2121}{\displaystyle\text{/}}} [/math]

[math]2\cdot \int_{-1}^{1}\sqrt{1-x^2}dx[/math]
or how about [math]\frac{ln(-1)}{i}[/math]
or one for engineers [math]\pi=3[/math]