pi-2-1-pi-2-3-pi-2-9pi-2-5-3pi-2-25pi-2-7-5pi-2-49pi-2-9-7pi-2-81pi-2-11-9pi-2-121pi-2-

Question Number 127682 by Dwaipayan Shikari last updated on 31/Dec/20

\frac{π^{2}}{1 + \frac{π^{2}}{3 - π^{2} + \frac{9 π^{2}}{5 - 3 π^{2} + \frac{25 π^{2}}{7 - 5 π^{2} + \frac{49 π^{2}}{9 - 7 π^{2} + \frac{81 π^{2}}{11 - 9 π^{2} + \frac{121 π^{2}}{\dots . .}}}}}}}

Commented by Dwaipayan Shikari last updated on 01/Jan/21

π {t a n}^{- 1} (π)

Commented by A8;15: last updated on 03/Jan/21

It will be nice if you show us a proof, sir!

Answered by Dwaipayan Shikari last updated on 03/Jan/21

{t a n}^{- 1} x = x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \dots .

= x + x (- \frac{x^{2}}{3}) + x (- \frac{x^{2}}{3}) (- \frac{3 x^{2}}{5}) + x (- \frac{x^{2}}{3}) (- \frac{3 x^{2}}{5}) (- \frac{5 x^{2}}{7}) + . .

= \frac{x}{1 + \frac{\frac{x^{2}}{3}}{1 - \frac{x^{2}}{3} + \frac{\frac{x^{2}}{5}}{1 - \frac{x^{2}}{5} + \frac{\frac{x^{2}}{7}}{1 - \frac{x^{2}}{7} + \dots}}}} = \frac{x}{1 + \frac{x^{2}}{3 - x^{2} + \frac{3^{2} x^{2}}{5 - 3 x^{2} + \frac{5^{2} x^{2}}{7 - 5 x^{2} + . .}}}}

T a k e x = π

Commented by Dwaipayan Shikari last updated on 03/Jan/21

https://en.wikipedia.org/wiki/Euler%27s_continued_fraction_formula

Commented by A8;15: last updated on 03/Jan/21

Thanks sir . My respect for you!

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