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Question Number 184341 by Davidtim last updated on 05/Jan/23 $${li}\underset{{x}\rightarrow\infty} {{m}}\frac{{x}^{\mathrm{2}} }{\mathrm{3}^{{x}} }=? \\ $$ Commented by JDamian last updated on 05/Jan/23 zero, as the denominator tends to infinity faster than the numerator does. Answered by…

Question Number 118791 by Algoritm last updated on 19/Oct/20 Answered by Olaf last updated on 19/Oct/20 $${f}_{{p}} \left({a}\right)\:=\:\underset{{n}=\mathrm{1}} {\overset{{p}} {\sum}}\frac{\mathrm{sin}\left(\mathrm{2}{n}\ast{a}\right)}{\mathrm{2}^{{n}} } \\ $$$${f}_{{p}} \left({a}\right)\:=\:\underset{{n}=\mathrm{1}} {\overset{{p}}…

Question Number 118781 by Algoritm last updated on 19/Oct/20 Commented by MJS_new last updated on 19/Oct/20 $$\mathrm{minimum}\:\mathrm{of}\:{f}\left({x}\right)={x}^{{x}^{{x}^{{x}} } } \:\mathrm{is}\:\approx.\mathrm{59} \\ $$$$\mathrm{but}\:\frac{\mathrm{1}}{\mathrm{3}^{\sqrt{\mathrm{48}}} }\approx.\mathrm{00049} \\ $$$$\Rightarrow\:\mathrm{no}\:\mathrm{solution}…

Question Number 118768 by mathdave last updated on 19/Oct/20 Answered by mindispower last updated on 23/Oct/20 $$\underset{{m}=\mathrm{0}} {\overset{{n}} {\sum}}{e}^{{imx}} =\frac{\mathrm{1}−\left({e}^{{ix}} \right)^{{n}+\mathrm{1}} }{\mathrm{1}−{e}^{{ix}} }=\frac{{e}^{{i}\frac{{nx}}{\mathrm{2}}} \left({e}^{−{i}\frac{\left({n}+\mathrm{1}\right){x}}{\mathrm{2}}} −{e}^{{i}\frac{\left({n}+\mathrm{1}\right){x}}{\mathrm{2}}}…

Question Number 118759 by ZiYangLee last updated on 19/Oct/20 $$\mathrm{The}\:\mathrm{first}\:\mathrm{three}\:\mathrm{terms}\:\mathrm{in}\:\mathrm{the}\:\mathrm{binomial}\:\mathrm{expansion} \\ $$$$\left({p}−{q}\right)^{{m}} \:,\:\mathrm{in}\:\mathrm{ascending}\:\mathrm{order}\:\mathrm{of}\:{q},\:\mathrm{are}\:\mathrm{denoted} \\ $$$$\mathrm{by}\:{a},{b}\:\mathrm{and}\:{c}\:\mathrm{respectively}. \\ $$$$\mathrm{Show}\:\mathrm{that}\:\frac{{b}^{\mathrm{2}} }{{ac}}=\frac{\mathrm{2}{m}}{{m}−\mathrm{1}} \\ $$ Commented by PRITHWISH SEN 2…