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Category: Differentiation

Advanced-Calculus-if-n-2-1-n-n-2-n-then-prove-that-1-2-e-1-proof-method-1-

Question Number 141668 by mnjuly1970 last updated on 22/May/21 $$\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:…….{Advanced}\:…\bigstar\:…\bigstar\:…\:{Calculus}……. \\ $$$$\:\:\:\:\:\:\:\:\:{if}\:\:\:\:\Omega\:=\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}} \zeta\left({n}\right)}{\mathrm{2}^{{n}} }\:{then}\:{prove} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:{that}\:::\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{2}}\:=\:{e}^{\Omega−\mathrm{1}} \:\: \\ $$$$\:\:\:\:{proof}\::: \\ $$$$\:\:\:\:{method}\:\left(\mathrm{1}\right):…

n-1-sin-2n-1-pi-6-2n-1-2-a-G-a-G-catalan-constant-

Question Number 141512 by mnjuly1970 last updated on 19/May/21 $$ \\ $$$$\:\:\:\underset{{n}=\mathrm{1}\:} {\overset{\infty} {\sum}}\frac{{sin}\left(\frac{\left(\mathrm{2}{n}−\mathrm{1}\right)\pi}{\mathrm{6}}\right)}{\left(\mathrm{2}{n}−\mathrm{1}\right)^{\mathrm{2}} }\:=\:{a}.{G} \\ $$$$\:\:{a}\:=\:?\:\:\:\:\:\:\left({G}:=\:{catalan}\:{constant}\right) \\ $$$$ \\ $$ Terms of Service Privacy…

prove-that-n-0-5n-2-5n-3-5n-1-5n-4-1-5-2-

Question Number 141378 by mnjuly1970 last updated on 18/May/21 $$\:\:\:{prove}\:{that}:: \\ $$$$\:\:\:\:\:\:\underset{{n}=\mathrm{0}} {\overset{\infty} {\prod}}\frac{\left(\mathrm{5}{n}+\mathrm{2}\right)\left(\mathrm{5}{n}+\mathrm{3}\right)}{\left(\mathrm{5}{n}+\mathrm{1}\right)\left(\mathrm{5}{n}+\mathrm{4}\right)}\:=\varphi\: \\ $$$$\:\:\:\:\:\:\:\varphi:=\:\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$ Answered by Dwaipayan Shikari last updated on…