Question Number 1067 by Yugi last updated on 30/May/15 $${Let}\: \\ $$$$\:\:\:\:\:\:{S}_{{N}} =\underset{{n}=\mathrm{1}} {\overset{{N}} {\sum}}\left(−\mathrm{1}\right)^{{n}−\mathrm{1}} {n}^{\mathrm{3}} \:. \\ $$$${Find}\:{S}_{\mathrm{2}{N}} \:{in}\:{terms}\:{of}\:{N}.\: \\ $$$$ \\ $$ Answered…
Question Number 1066 by 123456 last updated on 27/May/15 $$\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\frac{{x}}{\mathrm{tan}\:{x}}{dx} \\ $$ Answered by malwaan last updated on 01/Jun/15 $$\frac{\pi}{\mathrm{2}}{log}\left(\mathrm{2}\right) \\ $$ Terms…
Question Number 132138 by benjo_mathlover last updated on 11/Feb/21 $$\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}}\right)^{\mathrm{n}} }{\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{n}}\right)^{\mathrm{n}} }\:−\:\mathrm{e}^{\mathrm{2}} \:\right)\mathrm{n}^{\mathrm{2}} =? \\ $$ Answered by EDWIN88 last updated on 11/Feb/21 $$\mathrm{L}=\mathrm{e}^{\mathrm{2}}…
Question Number 66601 by aliesam last updated on 17/Aug/19 Commented by kaivan.ahmadi last updated on 17/Aug/19 $$\mathrm{14} \\ $$$${D}=\left(\mathrm{0},+\infty\right)\Rightarrow{c}\:{is}\:{true} \\ $$ Answered by Rio Michael…
Question Number 1063 by lops55 last updated on 26/May/15 $${rational}\:{number}\:{which}\:{is}\:{neither}\:{negetive}\:{nor}\:{positive} \\ $$ Answered by 123456 last updated on 01/Jun/15 $$\mathrm{0}? \\ $$ Terms of Service…
Question Number 66599 by aliesam last updated on 17/Aug/19 Commented by kaivan.ahmadi last updated on 17/Aug/19 $$\left({a}−{b}\right)^{\mathrm{2}} ={a}^{\mathrm{2}} −{b}^{\mathrm{2}} \Rightarrow{a}^{\mathrm{2}} −\mathrm{2}{ab}+{b}^{\mathrm{2}} ={a}^{\mathrm{2}} −{b}^{\mathrm{2}} \Rightarrow \\…
Question Number 132135 by benjo_mathlover last updated on 11/Feb/21 Answered by Olaf last updated on 11/Feb/21 $${f}\left({x}\right)+{f}\left(\frac{{x}−\mathrm{1}}{{x}}\right)\:=\:\mathrm{1}+{x}\:\:\:\:\:\left(\mathrm{1}\right) \\ $$$$\mathrm{Let}\:{x}\:=\:\frac{{u}−\mathrm{1}}{{u}} \\ $$$$\left(\mathrm{1}\right)\::\:{f}\left(\frac{{u}−\mathrm{1}}{{u}}\right)+{f}\left(\frac{\frac{{u}−\mathrm{1}}{{u}}−\mathrm{1}}{\frac{{u}−\mathrm{1}}{{u}}}\right)\:=\:\mathrm{1}+\frac{{u}−\mathrm{1}}{{u}} \\ $$$${f}\left(\frac{{u}−\mathrm{1}}{{u}}\right)+{f}\left(\frac{\mathrm{1}}{\mathrm{1}−{u}}\right)\:=\:\mathrm{1}+\frac{{u}−\mathrm{1}}{{u}}\:\:\:\:\:\left(\mathrm{2}\right) \\ $$$$\left(\mathrm{1}\right)−\left(\mathrm{2}\right)\::…
Question Number 132131 by Ar Brandon last updated on 11/Feb/21 Commented by Ar Brandon last updated on 11/Feb/21 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{relation}\:\mathrm{between}\:\mathrm{the}\:\mathrm{areas}\:\mathrm{of}\: \\ $$$$\mathrm{triangle}\:\mathrm{ABC}\:\mathrm{and}\:\mathrm{triangle}\:\mathrm{A}'\mathrm{B}'\mathrm{C}' \\ $$ Commented by…
Question Number 1057 by 123456 last updated on 25/May/15 $${f}:\mathbb{R}_{+} \rightarrow\mathbb{R} \\ $$$${x}={i}+{j} \\ $$$${x}\in\mathbb{R}_{+} \\ $$$${i}\in\mathbb{N} \\ $$$${j}\in\left[\mathrm{0},\mathrm{1}\right) \\ $$$${f}\left({x}\right)=\begin{cases}{{f}\left({i}−\mathrm{1}\right)+\left({i}+\mathrm{1}\right)\left({j}+\mathrm{1}\right)}&{{x}\geqslant\mathrm{1}}\\{{j}}&{\mathrm{0}\leqslant{x}<\mathrm{1}}\\{{x}}&{{x}<\mathrm{0}}\end{cases} \\ $$$${f}\left(\mathrm{9}.\mathrm{5}\right)=? \\ $$…
Question Number 1056 by 123456 last updated on 25/May/15 $${y}''+{y}=\mathrm{8}{cos}\left({t}\right) \\ $$$${y}\left(\mathrm{0}\right)=+\mathrm{1} \\ $$$${y}'\left(\mathrm{0}\right)=−\mathrm{1} \\ $$ Commented by prakash jain last updated on 26/May/15 $$\mathrm{characterstic}\:\mathrm{equation}…