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I'm new to math.stackexchange so apologies in advance for any blunders:

I am trying to calculate $\pi$ using the following technique here.

Considering the above link says:

The discovery of this formula came as a surprise. For centuries it had been assumed that there was no way to compute the $n$th digit of $\pi$ without calculating all of the preceding $n − 1$ digits.

I won't pretend I understand the calculation and may have misunderstood, but from what I can gather, this formula should identify the $k$th digit of $\pi$ independently (ie without having to calculate the previous $k-1$ digits):

$$ \sum_{k=0}^\infty \left[ { 1 \over 16^k} \left( {120k^2 + 151k + 47 \over 512k^4 + 1024k^3 + 712k^2 + 194k + 15 } \right) \right]$$

The values I'm getting out are:

k  result
0  3.133333
1  0.0080891331
2  0.0001649239

....I realise that summing these gives me the correct digits of $\pi$ to an accuracy of $k$, however by calculating each of these in turn it seem that we are still "calculating all of the preceding n-1 digits"?

Michael Hardy
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Bendy
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1 Answers1

2

This is not for decimal digits, it is for hexadecimal digits. See this part of that Wikipedia page for the procedure.

There is nothing as good (in terms of time) for decimal digits, but Fabrice Bellard has an $O(n^2)$ method.

Robert Israel
  • 448,999