Question Number 4584 by MrGreg last updated on 08/Feb/16 $$ \\ $$ Answered by FilupSmith last updated on 09/Feb/16 $$\:\:\:\:{S}=\mathrm{1}−\mathrm{2}+\mathrm{4}−\mathrm{8}+\mathrm{16}−\mathrm{32}+\mathrm{64}−… \\ $$$$+{S}=\mathrm{0}+\mathrm{1}−\mathrm{2}+\mathrm{4}−\mathrm{8}+\mathrm{16}−\mathrm{32}+\mathrm{64}−… \\ $$$$\mathrm{2}{S}=\mathrm{1}+\left(−\mathrm{1}+\mathrm{2}−\mathrm{4}+\mathrm{8}−\mathrm{16}+…\right) \\…
Question Number 135638 by Dwaipayan Shikari last updated on 14/Mar/21 $$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\sqrt{\mathrm{5}}−\mathrm{2}\right)^{\mathrm{2}{n}+\mathrm{1}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} }=\frac{\pi^{\mathrm{2}} }{\mathrm{24}}−\frac{\mathrm{1}}{\mathrm{12}}{log}^{\mathrm{2}} \left(\mathrm{2}+\sqrt{\mathrm{5}}\right) \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 70069 by tw000001 last updated on 01/Oct/19 $$\underset{{n}=\mathrm{1}} {\overset{\mathrm{5}} {\prod}}\frac{\left(\mathrm{12}{n}−\mathrm{2}\right)^{\mathrm{4}} +\mathrm{18}^{\mathrm{2}} }{\left(\mathrm{12}{n}−\mathrm{8}\right)^{\mathrm{4}} +\mathrm{18}^{\mathrm{2}} } \\ $$$$=\frac{\left(\mathrm{10}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{22}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{34}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{46}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{58}^{\mathrm{4}} +\mathrm{324}\right)}{\left(\mathrm{4}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{16}^{\mathrm{4}} +\mathrm{324}\right)\left(\mathrm{28}^{\mathrm{4}}…
Question Number 70051 by Aditya789 last updated on 30/Sep/19 $$\frac{{a}+{b}}{{c}}=\frac{{cos}\left(\frac{{a}−{b}}{\mathrm{2}}\right)}{{cos}\frac{{c}}{\mathrm{2}}} \\ $$ Answered by $@ty@m123 last updated on 01/Oct/19 $${LHS}=\frac{{a}+{b}}{{c}}=\frac{\mathrm{sin}\:{A}+\mathrm{sin}\:{B}}{\mathrm{sin}\:{C}} \\ $$$$=\frac{\mathrm{2sin}\:\frac{{A}+{B}}{\mathrm{2}}\mathrm{cos}\:\frac{{A}−{B}}{\mathrm{2}}}{\mathrm{2sin}\:\frac{{C}}{\mathrm{2}}\mathrm{cos}\:\frac{{C}}{\mathrm{2}}} \\ $$$$=\frac{\mathrm{sin}\:\frac{\pi−{C}}{\mathrm{2}}\mathrm{cos}\:\frac{{A}−{B}}{\mathrm{2}}}{\mathrm{sin}\:\frac{{C}}{\mathrm{2}}\mathrm{cos}\:\frac{{C}}{\mathrm{2}}} \\…
Question Number 4511 by madscientist last updated on 04/Feb/16 $${is}\:{this}\:{true}? \\ $$$$\int_{\mathrm{0}} ^{\infty} {e}^{−{t}\:} {dt}=\mathrm{1}\: \\ $$$${if}\:{so}\:{how}? \\ $$$$ \\ $$ Commented by FilupSmith last…
Question Number 4507 by 123456 last updated on 03/Feb/16 $$\mathrm{lets}\:{a}\:\mathrm{and}\:{b}\:\mathrm{be}\:\mathrm{two}\:\mathrm{sequence}\:\mathrm{such}\:\mathrm{that} \\ $$$${A}=\underset{{n}\rightarrow+\infty} {\mathrm{lim}}{a}_{{n}} \\ $$$${B}=\underset{{n}\rightarrow+\infty} {\mathrm{lim}}{b}_{{n}} \\ $$$$\mathrm{exist}\:\mathrm{and}\:\mathrm{are}\:\mathrm{finite},\:\mathrm{lets} \\ $$$${c}\:\mathrm{be}\:\mathrm{a}\:\mathrm{new}\:\mathrm{sequence} \\ $$$${c}_{{n}} ={p}\left({n}\right){a}_{\sigma\left({n}\right)} +{q}\left({n}\right){b}_{\mu\left({n}\right)} \\…
Question Number 70035 by TawaTawa last updated on 30/Sep/19 Commented by TawaTawa last updated on 30/Sep/19 Commented by mr W last updated on 30/Sep/19 $${let}\:{x}={XP}…
Question Number 70030 by Joel122 last updated on 30/Sep/19 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{convergence}\:\mathrm{of} \\ $$$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\frac{\mathrm{1}}{{n}}\:+\:\mathrm{1}}{−{n}^{\mathrm{2}} } \\ $$ Answered by mind is power last updated on…
Question Number 70022 by naka3546 last updated on 30/Sep/19 $${Find}\:\:\:{all}\:\:{pairs}\:\:{of}\:\:\:\left({p},\:{q}\right)\:\:{integer}\left({s}\right)\:\:{such}\:\:{that} \\ $$$${p}^{\mathrm{3}} \:−\:{q}^{\mathrm{5}} \:\:=\:\:\left({p}\:+\:{q}\right)^{\mathrm{2}} \\ $$ Commented by MJS last updated on 30/Sep/19 $${p}=\mathrm{0}\:{q}=\mathrm{0} \\…
Question Number 70017 by Shamim last updated on 30/Sep/19 $$\mathrm{Solution}- \\ $$$$\mathrm{log}_{\mathrm{8}} \mathrm{x}+\mathrm{log}_{\mathrm{4}} \mathrm{x}+\mathrm{log}_{\mathrm{2}} \mathrm{x}=\mathrm{11} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{x}} \mathrm{8}}+\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{x}} \mathrm{4}}+\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{x}} \mathrm{2}}=\mathrm{11} \\ $$$$\Rightarrow\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{x}} \mathrm{2}^{\mathrm{3}} }+\frac{\mathrm{1}}{\mathrm{log}_{\mathrm{x}} \mathrm{2}^{\mathrm{2}}…