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Question Number 118791 by Algoritm last updated on 19/Oct/20 Answered by Olaf last updated on 19/Oct/20 $$f_p\left(a\right)=\underset{n=1}{\sum ^p}\frac{\sin (2n\ast a)}{2^n}$$$${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 $$\mathrm{minimum}\mathrm{of}f\left(x\right)=x^{x^{x^x}}\mathrm{is}\approx .59$$$$\mathrm{but}\frac{1}{3^{\sqrt{48}}}\approx .00049$$$$\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 $$\mathrm{The}\mathrm{first}\mathrm{three}\mathrm{terms}\mathrm{in}\mathrm{the}\mathrm{binomial}\mathrm{expansion}$$$${(p-q)}^m,\mathrm{in}\mathrm{ascending}\mathrm{order}\mathrm{of}q,\mathrm{are}\mathrm{denoted}$$$$\mathrm{by}a,b\mathrm{and}c\mathrm{respectively}.$$$$\mathrm{Show}\mathrm{that}\frac{b^2}{ac}=\frac{2m}{m-1}$$ Commented by PRITHWISH SEN 2…

Question Number 118733 by mohammad17 last updated on 19/Oct/20 Commented by bemath last updated on 19/Oct/20 $$\left(4\right)\frac{d}{dx}[\int {}_{x^3}^{2x^2}\cos (2t^3+1)dt]=$$$$\mathrm{4}{x}\:\mathrm{cos}\:\left(\mathrm{2}.\left(\mathrm{8}{x}^{\mathrm{6}} \right)+\mathrm{1}\right)−\mathrm{3}{x}^{\mathrm{2}}…