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Author: Tinku Tara

Prove-that-m-Z-r-1-m-x-r-1-m-m-1-2-mx-2pi-m-1-2-mx-x-0-t-x-1-e-t-dt-x-gt-0-

Question Number 2340 by Yozzi last updated on 18/Nov/15 $${Prove}\:{that},\:\forall{m}\in\mathbb{Z}^{+} , \\ $$$$\underset{{r}=\mathrm{1}} {\overset{{m}} {\prod}}\Gamma\left({x}+\frac{{r}−\mathrm{1}}{{m}}\right)={m}^{\frac{\mathrm{1}}{\mathrm{2}}−{mx}} \left(\mathrm{2}\pi\right)^{\frac{{m}−\mathrm{1}}{\mathrm{2}}} \Gamma\left({mx}\right). \\ $$$$\left\{\Gamma\left({x}\right)=\int_{\mathrm{0}} ^{\infty} {t}^{{x}−\mathrm{1}} {e}^{−{t}} {dt},\:{x}>\mathrm{0}\right\} \\ $$…

dx-x-2-3x-9-x-2-5x-7-

Question Number 2334 by Syaka last updated on 16/Nov/15 $$\int\frac{{dx}}{\left({x}^{\mathrm{2}} \:+\:\mathrm{3}{x}\:+\:\mathrm{9}\right)\sqrt{{x}^{\mathrm{2}} \:+\:\mathrm{5}{x}\:+\:\mathrm{7}}\:}\:\:=\:? \\ $$ Commented by prakash jain last updated on 16/Nov/15 $${x}^{\mathrm{2}} +\mathrm{5}{x}+\mathrm{7}=\left({x}+\frac{\mathrm{5}}{\mathrm{2}}\right)^{\mathrm{2}} +\frac{\mathrm{3}}{\mathrm{4}}…

verify-that-x-and-x-are-the-solution-of-the-homogeneous-equation-corresponding-to-1-x-y-2-xy-1-y-2-x-1-2-x-0-lt-x-lt-1-and-find-the-general-solution-

Question Number 133403 by Engr_Jidda last updated on 21/Feb/21 $${verify}\:{that}\:\varrho^{{x}} \:{and}\:{x}\:{are}\:{the}\:{solution} \\ $$$${of}\:{the}\:{homogeneous}\:{equation}\:{corresponding} \\ $$$${to}\:\left(\mathrm{1}−{x}\right){y}^{\mathrm{2}} +{xy}^{\mathrm{1}} −{y}=\mathrm{2}\left({x}−\mathrm{1}\right)^{\mathrm{2}} \varrho^{{x}\:} ,\:\mathrm{0}<{x}<\mathrm{1} \\ $$$${and}\:{find}\:{the}\:{general}\:{solution}. \\ $$ Terms of…

a-n-1-n-n-9-1-n-0-a-n-b-n-a-n-1-9-a-n-n-1-n-0-b-n-

Question Number 2329 by 123456 last updated on 16/Nov/15 $${a}_{{n}} =\left(−\mathrm{1}\right)^{{n}} \left(\lfloor\frac{{n}}{\mathrm{9}}\rfloor+\mathrm{1}\right) \\ $$$$\underset{{n}\geqslant\mathrm{0}} {\sum}{a}_{{n}} =? \\ $$$${b}_{{n}} =\frac{\left({a}_{{n}} −\mathrm{1}\right)\left(\mathrm{9}−{a}_{{n}} \right)}{{n}+\mathrm{1}} \\ $$$$\underset{{n}\geqslant\mathrm{0}} {\sum}{b}_{{n}} =?…

1-2-1-2-2-1-2-2-2-

Question Number 2326 by Filup last updated on 16/Nov/15 $$\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}×\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}+\sqrt{\mathrm{2}}}}×\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\sqrt{\mathrm{2}}}}}×…=? \\ $$ Commented by Rasheed Soomro last updated on 16/Nov/15 $$\mathcal{N}{ice}\:\mathcal{A}{pproach}! \\ $$ Commented by…

Question-133398

Question Number 133398 by rexford last updated on 21/Feb/21 Answered by EDWIN88 last updated on 22/Feb/21 $$\overset{\rightarrow} {{a}}.\overset{\rightarrow} {{b}}+\overset{\rightarrow} {{b}}.\overset{\rightarrow} {{c}}=\left(\overset{\rightarrow} {{a}}+\overset{\rightarrow} {{c}}\right).\overset{\rightarrow} {{b}}=−\overset{\rightarrow} {{b}}.\overset{\rightarrow}…