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Question Number 133653 by greg_ed last updated on 23/Feb/21

$$\mathbf{hi},\mathbf{everybody}!$$$$\mathbf{how}\mathbf{to}\mathbf{prove}\mathbf{that}\mathit{\pi }\mathbf{is}\mathbf{an}\mathbf{irrational}\mathbf{number}???$$

Answered by Dwaipayan Shikari last updated on 23/Feb/21

$$1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\frac{1}{11}+...=\frac{\pi }{4}$$$$AsItisanInfinteseriessoitcanneverberational$$$$Otherseries$$$$1+\frac{1}{2^4}+\frac{1}{3^4}+...=\frac{{\pi }^4}{90}$$$$1+\frac{1}{2}{\left(\frac{1}{2.1!}\right)}^2+\frac{1}{3}{\left(\frac{1.3}{2^2.2!}\right)}^2+\frac{1}{4}{\left(\frac{1.3.5}{2^3.3!}\right)}^2+...=\frac{4}{\pi }$$

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