Getting closer to $\pi$ | Colin

A lovely curiosity came my way via @mikeandallie and @divbyzero:
In 1992 Daniel Shanks observed that if p~pi to n digits, then p+sin(p)~pi to 3n digits. For instance, 3.14+sin(3.14)=3.1415926529…

— Dave Richeson (@divbyzero) July 15, 2016

Isn’t that neat? If I use an estimate $p = 3.142$, then this method gives $\pi \approx p + \sin(p) = 3.141\ 592\ 653\ 6$, which is off by about $10^{-12}$ — even better than Shanks suggests.

So, why does it work?

It’s a two-step chain of reasoning: a trig identity and a Maclaurin series.

The trig identity is that $\sin(\pi – p) = \sin(p)$, by…

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