It is a celebrated equation that $$\frac{\pi}{4}=\cfrac{1}{1+\cfrac{1^2}{3+\cfrac{2^2}{5+\cfrac{3^2}{7+\ddots}}}}$$
However, there are two other conjectured equations that I found which, if true (they seem to be), might reveal a pattern.
$$\frac{\pi^2}{12}=\cfrac{1}{1+\cfrac{1^4}{3+\cfrac{2^4}{5+\cfrac{3^4}{7+\ddots}}}}$$
$$\frac{\pi^3}{36}=\cfrac{1}{1+\cfrac{1^6}{3+\cfrac{2^6}{5+\cfrac{3^6}{7+\ddots}}}}$$
Conjectured General Formula: For natural $n\geqslant 1$, $$\frac{\pi^n}{4\cdot 3^{n-1}}=\cfrac{1}{1+\cfrac{1^{2n}}{3+\cfrac{2^{2n}}{5+\cfrac{3^{2n}}{7+\ddots}}}}$$
Can these be numerically verified? I have not the skill to by-hand prove/disprove these, and have only been using Wolfram Alpha to arrive at these conjectures.
It would also be much appreciated if one could suggest a program I could install in order to evaluate these continued fractions independently, as well as the code required. Will PARI/GP suffice?
Thanks.
from Hot Weekly Questions - Mathematics Stack Exchange
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