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"?