I'm interested in generating digits of $\pi$ (I'm programming it in Python) and from my research it seems Chudnovsky algorithm is the fastest. Unfortunately for me, the Wikipedia page only really gives a massive equation with lots of symbols.
Edit: What would it look like as an infinite series?
I'm in year 11 so simpler explanations would be appreciated.
Best Answer
The Chudnovsky series is based on a hypergeometric series, which may be why you think it is expressible as a simple geometric series. However, in general hypergeometric series are not expressible as geometric series.
However, using the expression you've linked to at wikipedia, you can write a trivial implementation in Python:
$$\frac{1}{\pi}=12\sum_{k=0}^{\infty}{\frac{(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+3/2}}}$$
So we have:
$$\pi\approx1/\left(12\sum_{k=0}^{x}{\frac{(-1)^{k}(6k)!(13591409+545140134k)}{(3k)!(k!)^{3}640320^{3k+3/2}}}\right)$$
For some large value of $x$. So we can write the following:
However, bear in mind it's been a while since I've done Python scripting, this script was made just based on documentation I could find on the Python site, so I'm unsure if the syntax/semantics are correct, but the concept is there.
Hope this helps.