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

$$\frac{1}{1+\frac{1}{2+\frac{3^2}{2+\frac{5^2}{2+\frac{7^2}{....}}}}}=\frac{\pi }{4}$$$$$$

Commented by rs4089 last updated on 21/Nov/20

$$startfrom1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-.....=\frac{\pi }{4}$$

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

$$\frac{1-\frac{\pi }{4}}{\frac{\pi }{4}}=\frac{\frac{1}{3}-\frac{1}{5}+\frac{1}{7}-\frac{1}{9}+...}{1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-..}=\frac{\frac{1}{3}-\frac{1}{5}+\frac{1}{7}-\frac{1}{9}+...}{2(\frac{1}{3}-\frac{1}{5}+\frac{1}{7}-\frac{1}{9}+..)+3(\frac{1}{5}-\frac{1}{7}+....)}$$$$=\frac{1}{2+3.\frac{\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+..}{\frac{1}{3}-\frac{1}{5}+\frac{1}{7}-\frac{1}{9}+...}}=\frac{1}{2+3^2.\frac{\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+..}{1-\frac{3}{5}+\frac{3}{7}-\frac{3}{9}+..}}$$$$=\frac{1}{2+3^2.\frac{\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+..}{2(\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11})+5(\frac{1}{7}-\frac{1}{9}+...)}}$$$$=\frac{1}{2+\frac{3^2}{2+5\frac{\frac{1}{7}-\frac{1}{9}+..}{\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+..}}}=\frac{1}{2+\frac{3^2}{2+\frac{5^2}{2+\frac{7^2}{...}}}}$$$$\frac{1-\frac{\pi }{4}}{\frac{\pi }{4}}=\frac{1}{2+\frac{3^2}{2+\frac{5^2}{2+\frac{7^2}{2+..}}}}\Rightarrow \frac{4}{\pi }=1+\frac{1}{2+\frac{3^2}{2+\frac{5^2}{2+..}}}\Rightarrow \frac{\pi }{4}=\frac{1}{1+\frac{1^2}{2+\frac{3^2}{2+\frac{5^2}{2+...}}}}$$

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