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Question Number 184969 by SEKRET last updated on 14/Jan/23

$$$$$$$$$$$$$$3+\frac{1}{6+\frac{3^2}{6+\frac{5^2}{6+\frac{7^2}{6+\frac{9^2}{6+......}}}}}=\mathit{\pi }$$$$\mathbf{provet}\mathbf{that}.$$$$$$$$$$$$$$

Commented by Frix last updated on 15/Jan/23

$$\mathrm{I}\mathrm{know}\mathrm{the}\mathrm{proof}\mathrm{for}\mathrm{this}\mathrm{one}$$$$\pi =\frac{4}{1+\frac{1^2}{2+\frac{3^2}{2+\frac{5^2}{2+\frac{7^2}{2+...}}}}}$$$$\mathrm{which}\mathrm{was}\mathrm{found}\mathrm{by}JohnWallis\mathrm{in}1665.$$

Commented by SEKRET last updated on 15/Jan/23

$$??$$

Answered by aba last updated on 14/Jan/23

$$\left[3\right]=3$$$$[3;6]=3+\frac{1}{6}=\frac{19}{6}=3,166...$$$$[3;6;6]=3+\frac{1}{6+\frac{9}{6}}=\frac{141}{45}=3,1333...$$$$[3;6;6;6]=3+\frac{1}{6+\frac{9}{6+\frac{25}{6}}}=\frac{1321}{420}=3,145...$$$$[3;6;6;6;6]=3+\frac{1}{6+\frac{9}{6+\frac{25}{6+\frac{49}{6}}}}=\frac{989}{315}=3,13968...$$$$[3;6;6;6;6;6]=3+\frac{1}{6+\frac{9}{6+\frac{25}{6+\frac{49}{6+\frac{121}{6}}}}}=\frac{248851}{79170}=3,143...$$$$[3;6;6;6;6..........\mathrm{\infty }]=\pi$$

Commented by Frix last updated on 15/Jan/23

$$\mathrm{This}\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{proof}.$$

Commented by aba last updated on 15/Jan/23

$$\mathrm{this}\mathrm{is}\mathrm{a}\mathrm{proposal}$$

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