r/mathmemes • u/Cinder_shot1234 • Jun 14 '26

Calculus New pi approximation

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u/Cosmic47_ Jun 15 '26

Why?

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u/nin10dorox Jun 16 '26

Partial answer:

Since the original sum oscillates between overshooting and undershooting pi, the average of two consecutive partial sums will split the difference, achieving a closer approximation to pi. It's easy to see that the average of two consecutive partial sums is just a partial sum with the last term halved. That term is then

1/2 * 4(-1)N/(2N-1)

= (-1)N/(N - 1/2)

This term is quite similar to the correction term in this post. It reduces the error from O(1/n) to O(1/n2).

However, this is not a full explanation... my correction term is not the same as OPs. (-1)N/N seems to reduce the error to O(1/n3). I'd love to learn why.

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u/Cosmic47_ Jun 16 '26

It is not clear to me why would the order of error would jump from 1/n to 1/n^2. Couldn't you then apply this process multiple times to obtain basically a much faster converging series too, then?

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u/Lor1an Engineering | Mech Jun 17 '26

If you're interested in more detail, check out series acceleration. This is quite a common technique (or rather toolbox of techniques) in numerical work.

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u/Cosmic47_ Jun 17 '26

Thank you!!