Is there a formula to approximate πin the form of p q ?

Question

Is there a formula to approximate   πin the form of             p      q      ?

Alessandro Schmidt · Accepted Answer

This series converges to   π from above, starting at       22    7    π  =      22    7    −      ∑          k      =      0        ∞              96      (      160              k        2            +      422      k      +      405      )              (      4      k      +      3      )      (      4      k      +      5      )      (      4      k      +      7      )      (      4      k      +      9      )      (      4      k      +      11      )      (      4      k      +      13      )      (      4      k      +      15      )      (      4      k      +      17      )      A simpler one is  π  =      22    7    −      ∑          k      =      1              ∞            240          (      4      k      +      1      )      (      4      k      +      2      )      (      4      k      +      3      )      (      4      k      +      5      )      (      4      k      +      6      )      (      4      k      +      7      )

bs1tuaz · Accepted Answer

Yes. There are several. The best know is the Leibniz formula:

1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots = \frac{π}{4}

This is an infinite series but just do the first few terms to get:

π = 4 \frac{3 \cdot 5 \cdot 7 - 5 \cdot 7 + 3 \cdot 7 - 3 \cdot 5}{3 \cdot 5 \cdot 17} = \frac{304}{105} \approx 2.8952380952380 \dots

Okay, I never said it would be a good approximation, but each successive term gets closer. Add

4 / 9

to that and we get

3.3396825396825 \dots

EDIT
Here's a better one:

\frac{π}{4} = \sum_{n = 0}^{\infty} \frac{2}{(4 n + 1) (4 n + 3)}

So for the first three terms we get:

π = 8 (\frac{1}{3} + 7 \frac{1}{5} + 11 \frac{1}{9} + 15 \frac{1}{13}) \approx 3.017071817071817 \dots

Is there a formula to approximate π<!-- π

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