Question Number 72020 by mathmax by abdo last updated on 23/Oct/19 $${calculate}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:{C}_{{n}} ^{{k}} \:{cos}\left(\frac{{k}\pi}{\mathrm{2}{n}}\right) \\ $$ Commented by mathmax by abdo last updated…
Question Number 72018 by mathmax by abdo last updated on 23/Oct/19 $${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{3}} \left({n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$ Commented by mathmax by abdo last…
Question Number 6482 by prakash jain last updated on 28/Jun/16 $$\mathrm{Is}\:\mathrm{there}\:{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{such}\:\mathrm{that}\:{f}\left({x}\right)\neq\mathrm{0} \\ $$$${f}\left({x}^{\mathrm{3}} \right)=\mathrm{3}{f}\left({x}^{\mathrm{2}} \right) \\ $$$$ \\ $$ Commented by Yozzii last updated on…
Question Number 137365 by liberty last updated on 02/Apr/21 $${Given}\:{f}\left({x}^{\mathrm{2}} +{x}\right)+\mathrm{2}{f}\left({x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}\right)=\:\mathrm{9}{x}^{\mathrm{2}} −\mathrm{15}{x} \\ $$$${find}\:{the}\:{value}\:{of}\:{f}\left(\mathrm{2017}\right). \\ $$ Answered by MJS_new last updated on 02/Apr/21 $${f}\left({t}\right)={at}+{b}…
Question Number 6268 by sanusihammed last updated on 21/Jun/16 $${Test}\:{for}\:{convergence}\:{of}\:{the}\:{series}\: \\ $$$$\infty \\ $$$$\Sigma\:\:\:\:\:\frac{{sin}\left({nx}\right)}{{n}^{\mathrm{2}} } \\ $$$${n}\:=\:\mathrm{1}\: \\ $$ Commented by FilupSmith last updated on…
Question Number 6267 by sanusihammed last updated on 21/Jun/16 $${Verify}\:{the}\:{convergence}\:{of}\:{the}\:{exponential}\:{series}\:\:{e}^{{x}} \\ $$$${using}\:{D}'{Alermbert}\:{ratio}\:{test}. \\ $$ Commented by Yozzii last updated on 21/Jun/16 $${u}_{{r}} =\frac{{x}^{{r}} }{{r}!}\Rightarrow\frac{{u}_{{r}+\mathrm{1}} }{{u}_{{r}}…
Question Number 137335 by liberty last updated on 01/Apr/21 $$ \\ $$P(x) = 3x^75 + 2x^14 – 3x^2 – 1. What is the remainder when…
Question Number 6238 by sanusihammed last updated on 19/Jun/16 Answered by Yozzii last updated on 20/Jun/16 $${Let}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{1}+{n}^{−\mathrm{1}} \right)^{−{n}^{\mathrm{2}} } ={l}. \\ $$$${Let}\:{u}={n}^{−\mathrm{1}} \Rightarrow{l}=\underset{{u}\rightarrow\mathrm{0}^{+} }…
Question Number 137282 by mathlove last updated on 31/Mar/21 Answered by EDWIN88 last updated on 31/Mar/21 $$\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{fofofof}\right)\left(\mathrm{x}\right)=\mathrm{f}\:'\left(\mathrm{x}\right)\mathrm{f}\:'\left(\mathrm{f}\left(\mathrm{x}\right)\right)\mathrm{f}\:'\left(\mathrm{f}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\right)\:\mathrm{f}\:'\left(\mathrm{f}\left(\mathrm{f}\left(\mathrm{f}\left(\mathrm{x}\right)\right)\right)\right) \\ $$ Answered by floor(10²Eta[1]) last updated on…
Question Number 137280 by mathlove last updated on 31/Mar/21 Answered by EDWIN88 last updated on 31/Mar/21 $$\Leftrightarrow\:\mathrm{y}\:=\:\mathrm{x}^{\left(\frac{\pi}{\mathrm{ln}\:\mathrm{x}}\right)} =\:\mathrm{x}^{\pi\mathrm{log}\:_{\mathrm{x}} \left(\mathrm{e}\right)} \\ $$$$\Leftrightarrow\:\mathrm{y}=\:\mathrm{x}^{\mathrm{log}\:_{\mathrm{x}} \left(\mathrm{e}^{\pi} \right)} \:;\:\mathrm{y}\:=\:\mathrm{e}^{\pi} \:\Rightarrow\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\mathrm{0}.…