1-1-1-2-3-2-2-5-2-2-7-2-pi-4-

Question Number 122960 by Dwaipayan Shikari last updated on 21/Nov/20

\frac{1}{1 + \frac{1}{2 + \frac{3^{2}}{2 + \frac{5^{2}}{2 + \frac{7^{2}}{\dots .}}}}} = \frac{π}{4}

Commented by rs4089 last updated on 21/Nov/20

s t a r t f r o m 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \dots . . = \frac{π}{4}

Commented by Dwaipayan Shikari last updated on 21/Nov/20

\frac{1 - \frac{π}{4}}{\frac{π}{4}} = \frac{\frac{1}{3} - \frac{1}{5} + \frac{1}{7} - \frac{1}{9} + \dots}{1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - . .} = \frac{\frac{1}{3} - \frac{1}{5} + \frac{1}{7} - \frac{1}{9} + \dots}{2 (\frac{1}{3} - \frac{1}{5} + \frac{1}{7} - \frac{1}{9} + . .) + 3 (\frac{1}{5} - \frac{1}{7} + \dots .)}

= \frac{1}{2 + 3 . \frac{\frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + . .}{\frac{1}{3} - \frac{1}{5} + \frac{1}{7} - \frac{1}{9} + \dots}} = \frac{1}{2 + 3^{2} . \frac{\frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + . .}{1 - \frac{3}{5} + \frac{3}{7} - \frac{3}{9} + . .}}

= \frac{1}{2 + 3^{2} . \frac{\frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + . .}{2 (\frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11}) + 5 (\frac{1}{7} - \frac{1}{9} + \dots)}}

= \frac{1}{2 + \frac{3^{2}}{2 + 5 \frac{\frac{1}{7} - \frac{1}{9} + . .}{\frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + . .}}} = \frac{1}{2 + \frac{3^{2}}{2 + \frac{5^{2}}{2 + \frac{7^{2}}{\dots}}}}

\frac{1 - \frac{π}{4}}{\frac{π}{4}} = \frac{1}{2 + \frac{3^{2}}{2 + \frac{5^{2}}{2 + \frac{7^{2}}{2 + . .}}}} \Rightarrow \frac{4}{π} = 1 + \frac{1}{2 + \frac{3^{2}}{2 + \frac{5^{2}}{2 + . .}}} \Rightarrow \frac{π}{4} = \frac{1}{1 + \frac{1^{2}}{2 + \frac{3^{2}}{2 + \frac{5^{2}}{2 + \dots}}}}