Extracting the nth digit of pi using Plouffe's formula?

Question

I have come upon the following formula to extract the nth digit of pi in base 10: $$\pi + 3 = \sum_{n=1}^{\infty} \frac{n 2^n n!^2}{(2n)!} $$ But this just seems to be a formula for pi. How can I use this formula to extract the nth digit of pi?

Calculate it out until the term is less than the signicance of the digt. — fleablood, Mar 17 '16 at 22:23
You use a spigot algorithm as per here. http://www.cut-the-knot.org/Curriculum/Algorithms/SpigotForPi.shtml — fleablood, Mar 18 '16 at 02:15

score 4 · Accepted Answer · answered Mar 23 '16 at 10:28

It is easier to find the $n$-th digit in base $d$ if you have a formula of the form

$$\sum_k \frac{\textbf{maybe large}}{\textbf{small} \times d^k},$$

like BBP formula for $d=16$. Just multiply the formula by $d^n$ and simplify the first $n$ summands as

$$\frac{(\textbf{maybe large} \times d^{n-k}) \text{ mod } \textbf{small}}{\textbf{small}}.$$

Then you need to sum up $n$ real numbers between $0$ and $1$ and several of the next terms (until the rest of series is small enough), see an example for $\ln 2$.

For BBP formula or $\ln 2$ example, the complexity of each step is $O(\log k)$, so the overall complexity is $O(n \log n)$. Your formula is used by Plouffe (difficult to read) with complexity $O(n^3 \log^3 n)$ -- but in any base. The version by Goudon is even more sophisticated, with complexity less than $O(n^2)$ in any base, but Proposition 1 from there shows that the idea is as in the answer.

Extracting the nth digit of pi using Plouffe's formula?

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