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Category: Integration

nice-calculus-prove-that-n-0-tan-1-1-F-n-tan-1-1-F-n-1-pi-2-4-F-n-is-fibonacci-sequence-

Question Number 137605 by mnjuly1970 last updated on 04/Apr/21 $$\:\:\:\:\:\:\:\:\:\:\:\:…..{nice}\:\:\:\:{calculus}… \\ $$$$\:\:\:\:\:{prove}\:{that}:: \\ $$$$\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{{F}_{{n}} }\right).{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{{F}_{{n}+\mathrm{1}} }\right)=\frac{\pi^{\mathrm{2}} }{\mathrm{4}} \\ $$$$\:\:{F}_{{n}} \:{is}\:{fibonacci}\:{sequence}…. \\…

mathematical-analysis-II-prove-that-0-1-1-1-x-ln-x-2-2x-1-1-x-x-2-n-1-1-n-2-2n-n-pi-2-18-

Question Number 137610 by mnjuly1970 last updated on 04/Apr/21 $$\:\:\:\:\:\:\:\:\:\:……..\:{mathematical}\:\:\:{analysis}\:\left({II}\right)…. \\ $$$$\:\:\:\:{prove}\:\:{that}\::: \\ $$$$\:\:\:\:\:\boldsymbol{\Omega}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\mathrm{1}}{\mathrm{1}+{x}}{ln}\left(\frac{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{1}}{\mathrm{1}+{x}+{x}^{\mathrm{2}} }\right)=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{2}} \begin{pmatrix}{\mathrm{2}{n}}\\{\:\:{n}}\end{pmatrix}}=\frac{\pi^{\mathrm{2}} }{\mathrm{18}}.. \\ $$ Terms…

advanced-calculus-n-1-n-n-I-havefound-pi-4-36-

Question Number 137592 by mnjuly1970 last updated on 04/Apr/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:……{advanced}…..{calculus}…. \\ $$$$\:\:\:\:\boldsymbol{\Omega}=\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\psi''\left({n}\right)}{{n}}=??? \\ $$$$\:{I}\:{havefound}\:::\:\:\Omega=−\frac{\pi^{\mathrm{4}} }{\mathrm{36}}\:\:…\:! \\ $$ Answered by Dwaipayan Shikari last updated…