#infiniteproduct

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• An interesting infinite product! \[\displaystyle\prod_{n=1}^\infty\coth^{(-1)^n}\left(\dfrac{\pi n}{2}\right)=\dfrac{\sqrt\pi}{\sqrt[4]2\sqrt{\varpi}}\] where \(\varpi=2.62205755\ldots\) is the lemniscate constant (the ratio of the perimete

  Pustam पुस्तम পুস্তম 𑂣𑂳𑂮𑂹𑂞𑂧 🇳🇵@[email protected]♥ 0↻ 12024-05-12 10:58 UTCView post →

• For any integer \(n\geq3\), \[\left\lfloor\dfrac{1}{1-\displaystyle\prod_{k=n}^\infty\left(1-\dfrac{1}{F_k}\right)}\right\rfloor=F_{n-2}\] where \(F_n\) is the \(n\)-th Fibonacci number. #FibonacciNumber #Fibonacci #FibonacciSequence #Fibon

  Pustam पुस्तम পুস্তম 𑂣𑂳𑂮𑂹𑂞𑂧 🇳🇵@[email protected]♥ 0↻ 02024-05-09 09:08 UTCView post →

• Interestingly, \(\pi^\text{th}\) root of \(4\) can be expressed as the following limit. \[\boxed{\displaystyle\sqrt[\pi]4=\lim_{n\to\infty}\prod_{k=0}^n\dfrac{\pi}{2\tan^{-1}(n+k)}}\] Proof: Let \(\Phi_n=\displaystyle\prod_{k=0}^n\dfrac{\pi

  Pustam पुस्तम পুস্তম 𑂣𑂳𑂮𑂹𑂞𑂧 🇳🇵@[email protected]♥ 0↻ 02024-04-12 13:08 UTCView post →