Pretty good approximation for Pi

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    Approximation Pi
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SUMMARY

The discussion reveals that \(\sqrt[5]{306}\) serves as an effective approximation for Pi, yielding a value of approximately 3.14155. By incrementally adding fractions such as \(1/51\) and \(1/12997\), the approximation improves to 3.1415925 and 3.141592653587, respectively, demonstrating rapid convergence to the actual value of Pi (3.141592653589). The user inquires about the existence of a similar series for \(\pi^5\) and seeks to understand the underlying pattern of the numbers used in the approximation.

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cuallito
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So [itex]\sqrt[5]{306}[/itex] is a pretty good approximation for Pi (=3.14155).

If you add 1/51, so that you have [itex]\sqrt[5]{306+1/51}[/itex] you get 3.1415925 (last digit is 6 for actual Pi.)

If you add 1/12997, [itex]\sqrt[5]{306+1/51+1/12997}[/itex] you get 3.141592653587 (vs 3.141592653589 for actual Pi.)

And so on. As you can see it converges pretty rapidly!

I was wondering if there was a similar series for Pi^5 that has already been discovered?
 
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Is there some pattern? I.e. why 306, and can I easily see what comes after 51, 12997, ... ? Because if they are just "random" numbers then this is nice, but it won't be more useful than just learning decimals of pi.
 

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