Question Number 193866 by liuxinnan last updated on 21/Jun/23 $${prove} \\ $$$$\:\:\underset{{i}=\mathrm{1}} {\overset{+\infty} {\sum}}\frac{\mathrm{1}}{{n}^{{i}} }=\frac{\mathrm{1}}{{n}−\mathrm{1}}\:\:\:\:\:\:{n}\in\mathbb{N}^{\ast} \\ $$$${and}\:{if}\:{n}>\mathrm{0}\wedge\:{n}\in\mathbb{R} \\ $$$${is}\:{it}\:{right}? \\ $$ Answered by AST last…
Question Number 193863 by horsebrand11 last updated on 21/Jun/23 $$\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\mathrm{x}^{\mathrm{2}} −\mathrm{cos}\:\left(\frac{\mathrm{x}}{\mathrm{1}−\mathrm{x}^{\mathrm{2}} }\right)}{\mathrm{x}^{\mathrm{4}} }\:=? \\ $$ Answered by MM42 last updated on 21/Jun/23 $${u}\rightarrow\mathrm{0}\Rightarrow\mathrm{1}−{cosu}\:\sim\:\frac{\mathrm{1}}{\mathrm{2}}{u}^{\mathrm{2}} \\…
Question Number 193821 by leandrosriv02 last updated on 20/Jun/23 Commented by talminator2856792 last updated on 21/Jun/23 $$\:\:−\left({a}\:+\:{b}\:−\:\mathrm{1}\right)\:+\:{a}\:+\:{b} \\ $$$$\:\:=\:{a}\:−{a}\:+\:{b}\:−{b}\:+\mathrm{1} \\ $$$$\:\:=\:\mathrm{0}\:+\:\mathrm{0}\:+\mathrm{1} \\ $$$$\:\:=\:\mathrm{1} \\ $$…
Question Number 193790 by pascal889 last updated on 20/Jun/23 Answered by Subhi last updated on 20/Jun/23 $$\:\mathrm{2}^{\mathrm{4}} \sqrt{{x}}\:+\frac{\mathrm{2}}{\:^{\mathrm{4}} \sqrt{{x}}}\:=\:\mathrm{5} \\ $$$$\sqrt{{x}}−\frac{\mathrm{5}}{\mathrm{2}}\:^{\mathrm{4}} \sqrt{{x}}\:+\mathrm{1}\:=\:\mathrm{0} \\ $$$$\:^{\mathrm{4}} \sqrt{{x}}\:=\:\frac{\frac{\mathrm{5}}{\mathrm{2}}\pm\sqrt{\frac{\mathrm{25}}{\mathrm{4}}−\mathrm{4}}}{\mathrm{2}}=\frac{\frac{\mathrm{5}}{\mathrm{2}}\pm\frac{\mathrm{3}}{\mathrm{2}}}{\mathrm{2}}=\:\frac{\mathrm{1}}{\mathrm{2}}\:{or}\:\mathrm{2}…
Question Number 193819 by SaRahAli last updated on 20/Jun/23 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}cos}\:\left(\frac{\frac{\Pi}{\mathrm{3}}+{x}}{{x}}\right) \\ $$ Answered by MM42 last updated on 20/Jun/23 $${not}\:{available} \\ $$ Terms of…
Question Number 193809 by byaw last updated on 20/Jun/23 A(-1, 2), B(3, 5) and C(4, 8) are the vertices of triangle ABC. Forces whose magnitudes…
Question Number 193804 by Mastermind last updated on 20/Jun/23 $$\mathrm{Ques}.\:\mathrm{6}\: \\ $$$$\:\:\:\:\:\mathrm{Let}\:\left(\mathrm{G},\:\ast\right)\:\mathrm{be}\:\mathrm{a}\:\mathrm{group}.\:\mathrm{and}\:\mathrm{let} \\ $$$$\mathrm{C}=\left\{\mathrm{c}\in\mathrm{G}\::\:\mathrm{c}\ast\mathrm{a}\:=\:\mathrm{a}\ast\mathrm{c}\:\forall\mathrm{a}\in\mathrm{G}\right\}.\:\mathrm{Prove} \\ $$$$\mathrm{that}\:\mathrm{C}\:\mathrm{is}\:\mathrm{subgroup}\:\mathrm{of}\:\mathrm{G}.\:\mathrm{hence}\:\mathrm{or}\: \\ $$$$\mathrm{otherwise}\:\mathrm{show}\:\mathrm{that}\:\mathrm{C}\:\mathrm{is}\:\mathrm{Abelian}. \\ $$$$ \\ $$$$\left[\mathrm{Note}\:\mathrm{C}\:\mathrm{is}\:\mathrm{called}\:\mathrm{the}\:\mathrm{center}\:\mathrm{of}\:\mathrm{group}\:\mathrm{G}\right] \\ $$$$ \\…
Question Number 193835 by Skabetix last updated on 20/Jun/23 Commented by a.lgnaoui last updated on 21/Jun/23 $$\:\mathrm{respecter}\:\mathrm{les}\:\mathrm{mesures}\:\mathrm{pour}\:\mathrm{avoir}\:\mathrm{une} \\ $$$$\mathrm{idee}\:\mathrm{de}\:\mathrm{la}\:\mathrm{question} \\ $$$${Je}\:{crois}\:{que}\:{c}\:{est}\:{la}\:{premuere}\:{fois}\:{que} \\ $$$${tu}\:{envoie}\:{une}\:{question}\:{au}\:{forum}. \\ $$$$\mathrm{l}\:\mathrm{inconu}\:\mathrm{dans}\:\mathrm{votre}\:\mathrm{question}\:\mathrm{est}\:\boldsymbol{\mathrm{DE}}…
Question Number 193803 by cortano12 last updated on 20/Jun/23 $$\:\:\:\:\:\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{11}−\mathrm{x}}\:\mathrm{cos}\:\left(\frac{\pi}{\mathrm{x}−\mathrm{2}}\right)}{\mathrm{cot}\:\left(\mathrm{x}−\mathrm{2}\right)}=? \\ $$ Answered by horsebrand11 last updated on 20/Jun/23 $$\:=\:\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\frac{\mathrm{3}\:\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}−\frac{\pi}{\mathrm{x}−\mathrm{2}}\right)}{\mathrm{cot}\:\left(\mathrm{x}−\mathrm{2}\right)} \\ $$$$\:=\:\mathrm{3}\:\underset{{x}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\mathrm{sin}\:\left(\mathrm{x}−\mathrm{2}\right)\:\mathrm{sin}\:\pi\left(\frac{\mathrm{x}−\mathrm{4}}{\mathrm{x}−\mathrm{2}}\right)…
Question Number 193796 by Rajpurohith last updated on 20/Jun/23 $${Let}\:{G}\:{be}\:{a}\:{finite}\:{group},{f}\:{be}\:{an}\:{automorphism}\:{of}\:{G} \\ $$$${such}\:{that}\:{f}\left({x}\right)={x}\:\Rightarrow{x}={e}\:. \\ $$$${Then}\:{prove}\:{that}, \\ $$$$\left(\boldsymbol{{i}}\right)\forall{g}\in{G},\:\exists{x}\in{G}\:{such}\:{that}\:{g}={x}^{−\mathrm{1}} {f}\left({x}\right). \\ $$$$\left(\boldsymbol{{ii}}\right){If}\:\forall{x}\in{G}\:,\:{f}\left({f}\left({x}\right)\right)={x}\:\Rightarrow\:{G}\:{is}\:{Abelian}. \\ $$$$ \\ $$ Terms of…