Question Number 7507 by Master Moon last updated on 01/Sep/16 $$\boldsymbol{{Given}}\:\boldsymbol{{r}},\:\boldsymbol{{t}},\:\boldsymbol{{j}}\:>\mathrm{0}\:\boldsymbol{{and}}\:\boldsymbol{{n}}\geqslant\mathrm{1};\:\boldsymbol{{Prove}}\:\boldsymbol{{that}} \\ $$$$\frac{\left[\underset{\boldsymbol{{p}}_{\mathrm{1}} =\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\left(\boldsymbol{{p}}_{\mathrm{1}} ^{\boldsymbol{{r}}} +\boldsymbol{{r}}\right)+\underset{\boldsymbol{{p}}_{\mathrm{2}} =\mathrm{1}} {\overset{\boldsymbol{{n}}} {\sum}}\left(\boldsymbol{{p}}_{\mathrm{2}} ^{\boldsymbol{{t}}} +\boldsymbol{{t}}\right)\right]^{\mathrm{2}} }{\boldsymbol{{n}}^{\mathrm{2}} \left[\left(\boldsymbol{{n}}!\right)^{\frac{\boldsymbol{{r}}+\boldsymbol{{t}}}{\boldsymbol{{n}}}}…
Question Number 138576 by liberty last updated on 15/Apr/21 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\mathrm{1}+\mathrm{cot}\:\mathrm{2}{x}\right)^{\mathrm{2tan}\:\mathrm{2}{x}} \:=?\: \\ $$ Answered by phanphuoc last updated on 15/Apr/21 $${li}\underset{{u}\left({x}\right)−>\mathrm{0}} {{m}}\left(\mathrm{1}+{u}\left({x}\right)\right)^{\mathrm{1}/{u}\left({x}\right)} ={e} \\…
Question Number 73043 by mathmax by abdo last updated on 05/Nov/19 $${prove}\:{that}\:{H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} }{{k}}×{C}_{{n}} ^{{k}} \\ $$$${H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}} \\ $$ Commented…
Question Number 7506 by gourav~ last updated on 01/Sep/16 $$\int\left\{\frac{\mathrm{1}−\sqrt{{x}}}{\mathrm{1}+\sqrt{{x}}}\right\}^{\frac{\mathrm{1}}{\mathrm{2}}} \frac{{dx}}{{x}}=? \\ $$$$ \\ $$$$ \\ $$ Answered by Yozzia last updated on 01/Sep/16 $${Let}\:{I}=\int\frac{\mathrm{1}}{{x}}\sqrt{\frac{\mathrm{1}−\sqrt{{x}}}{\mathrm{1}+\sqrt{{x}}}}{dx}.…
Question Number 138579 by Jamshidbek last updated on 15/Apr/21 Answered by MJS_new last updated on 16/Apr/21 $${x}\approx\mathrm{5}.\mathrm{40343170} \\ $$ Commented by MJS_new last updated on…
Question Number 73040 by mathmax by abdo last updated on 05/Nov/19 $${prove}\:{that}\:\:\forall\left({n},{p}\right)\in{N}^{\bigstar} ×{N} \\ $$$$\left.\mathrm{1}\right)\sum_{{k}=\mathrm{0}} ^{{p}} \:\left(−\mathrm{1}\right)^{{k}} \:{C}_{{n}} ^{{k}} \:=\left(−\mathrm{1}\right)^{{p}} \:{C}_{{n}−\mathrm{1}} ^{{p}} \\ $$$$\left.\mathrm{2}\right)\forall\left({p},{q}\right)\in{N}^{\mathrm{2}} \:\:\:\:\sum_{{k}=\mathrm{0}}…
Question Number 73041 by mathmax by abdo last updated on 05/Nov/19 $${prove}\:{that}\:\forall{n}\in\:{N}\:\:\sum_{{k}=\mathrm{0}} ^{\mathrm{2}{n}} \:\left(−\mathrm{1}\right)^{{k}} \:\left({C}_{\mathrm{2}{n}} ^{{k}} \right)^{\mathrm{2}} \:=\left(−\mathrm{1}\right)^{{n}} \:{C}_{\mathrm{2}{n}} ^{{n}} \\ $$ Commented by mathmax…
Question Number 73039 by mathmax by abdo last updated on 05/Nov/19 $${let}\:{U}_{{n}} =\frac{{n}}{\mathrm{2}}\:{if}\:{n}\:{even}\:{and}\:{U}_{{n}} =\frac{{n}−\mathrm{1}}{\mathrm{2}}\:{if}\:{n}\:{odd}\:{let}\:{f}\left({n}\right)=\sum_{{k}=\mathrm{0}} ^{{n}} {U}_{{k}} \\ $$$${prove}\:{that}\:\forall\left({x},{y}\right)\in{N}^{\mathrm{2}} \:\:\:\:{f}\left({x}+{y}\right)−{f}\left({x}−{y}\right)={xy} \\ $$ Answered by mind is…
Question Number 138575 by KwesiDerek last updated on 15/Apr/21 $$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{x}} \\ $$$$\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\mathrm{2}^{\boldsymbol{\mathrm{x}}} =\mathrm{1} \\ $$$$\boldsymbol{\mathrm{Any}}\:\boldsymbol{\mathrm{help}} \\ $$ Commented by soudo last updated on 15/Apr/21…
Question Number 73036 by mathmax by abdo last updated on 05/Nov/19 $$\left.{calculate}\:\mathrm{1}\right)\:\sum_{{k}=\mathrm{1}} ^{{n}} \:{k}^{\mathrm{2}} \left({n}+\mathrm{1}−{k}\right) \\ $$$$\left.\mathrm{2}\right)\sum_{\mathrm{1}\leqslant{i}\leqslant{j}\leqslant{n}} \:{ij} \\ $$ Commented by mathmax by abdo…