Question Number 6599 by WAI LIN last updated on 05/Jul/16 $${Prove}\:{that}\:\underset{{j}=\mathrm{1}} {\overset{{N}} {\sum}}\left({X}_{{j}} −\mathrm{1}\right)^{\mathrm{2}} \:=\underset{{j}=\mathrm{1}} {\overset{{N}} {\sum}}\:{X}_{{j}} ^{\mathrm{2}} \:−\:\mathrm{2}\:\underset{{j}=\mathrm{1}} {\overset{{N}} {\sum}}\:{X}_{{j}} \:+\:{N}. \\ $$ Answered…
Question Number 137668 by mnjuly1970 last updated on 05/Apr/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:…….{advanced}\:…\:…\:…\:…\:{calculus}…. \\ $$$$\:\:{prove}\:{that}:::::\:\ast\ast\ast\ast\ast \\ $$$$\:\:\:\:\boldsymbol{\Omega}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\Gamma\left({n}+\mathrm{1}\right)\Gamma\left({x}+\mathrm{1}\right)}{\left({n}+\mathrm{1}\right)\Gamma\left({n}+{x}+\mathrm{2}\right)}=\psi'\left({x}+\mathrm{1}\right) \\ $$$${proof}:: \\ $$$$\:\:\:\:\Omega=\underset{{n}=\mathrm{0}\:} {\overset{\infty} {\sum}}\frac{\beta\:\left({n}+\mathrm{1},{x}+\mathrm{1}\right)}{{n}+\mathrm{1}}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left\{\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)}\int_{\mathrm{0}} ^{\:\mathrm{1}}…
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Question Number 137667 by aurpeyz last updated on 05/Apr/21 $${y}={log}_{\mathrm{2}} \left({x}^{\mathrm{5}} +\mathrm{4}\right)\:{find}\:\frac{{dy}}{{dx}} \\ $$ Answered by Dwaipayan Shikari last updated on 05/Apr/21 $${y}=\frac{{log}\left({x}^{\mathrm{5}} +\mathrm{4}\right)}{{log}\left(\mathrm{2}\right)}\Rightarrow\frac{{dy}}{{dx}}=\frac{\mathrm{1}}{{log}\left(\mathrm{2}\right)}.\frac{\mathrm{5}{x}^{\mathrm{4}} }{{x}^{\mathrm{5}}…
Question Number 6593 by Tawakalitu. last updated on 04/Jul/16 Commented by Yozzii last updated on 05/Jul/16 $${Given}\:{that}\:{p}\:{is}\:{prime}\:{and}\:{p}\leqslant\mathrm{100}, \\ $$$${show}\:{that}\:{there}\:{are}\:{solutions}\:\left({x},{y}\right),\: \\ $$$${x},{y}\in\mathbb{Z},\:{for}\:{which}\:{y}^{\mathrm{37}} \equiv{x}^{\mathrm{3}} +\mathrm{37}\:\left({mod}\:{p}\right). \\ $$$$−−−−−−−−−−−−−−−−−−−−−−−…
Question Number 6590 by Temp last updated on 04/Jul/16 $$\mathrm{Solve}\:\mathrm{for}\:{a} \\ $$$$\frac{{a}^{\mathrm{2}} +\mathrm{1}}{{a}}=\frac{{a}}{\mathrm{1}−{a}^{\mathrm{2}} } \\ $$ Commented by nburiburu last updated on 04/Jul/16 $$\mathrm{1}−{a}^{\mathrm{4}} ={a}^{\mathrm{2}}…
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Question Number 6588 by Temp last updated on 04/Jul/16 $$\mathrm{Expansion}\:\mathrm{of}\:\mathrm{Q6582} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} {e}^{−{ix}^{\mathrm{2}} } {dx}=??? \\ $$ Commented by FilupSmith last updated on 07/Jul/16…
Question Number 137656 by rs4089 last updated on 05/Apr/21 $$\int_{\mathrm{0}} ^{\infty} {sin}\left({x}^{{p}} \right){dx} \\ $$ Answered by Dwaipayan Shikari last updated on 05/Apr/21 $$\int_{\mathrm{0}} ^{\infty}…
Question Number 137659 by JulioCesar last updated on 05/Apr/21 Commented by JulioCesar last updated on 05/Apr/21 $${help}\:{me} \\ $$ Commented by MJS_new last updated on…