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Question Number 137347 by Dwaipayan Shikari last updated on 01/Apr/21

$$\frac{1}{1+\frac{{\pi }^2}{1+\frac{{(\pi +1)}^2}{1+\frac{{(\pi +2)}^2}{1+...}}}}+\frac{1}{1+\frac{{(\pi +1)}^2}{1+\frac{{(\pi +2)}^2}{1+...}}}=\frac{1}{\pi }$$$$\mathbf{Prove}\mathbf{or}\mathbf{disprove}$$

Answered by MJS_new last updated on 01/Apr/21

$$t=1+\frac{{(\pi +1)}^2}{1+\frac{{(\pi +2)}^2}{1+...}}\in \mathbb{R}\left[\mathrm{obviously}\right]$$$$\frac{1}{1+\frac{{\pi }^2}{t}}+\frac{1}{t}=\frac{1}{\pi }$$$$\mathrm{but}\mathrm{this}\mathrm{has}\mathrm{no}\mathrm{real}\mathrm{solution}\mathrm{for}t\Rightarrow \mathrm{wrong}$$

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

$$Thankssir!$$

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