Question Number 226509 by hardmath last updated on 01/Dec/25 $$\mathrm{Find}:\:\:\:\underset{\boldsymbol{\mathrm{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\mathrm{n}\centerdot\left(\mathrm{2n}\:+\:\mathrm{1}\right)^{\mathrm{2}} }\:=\:? \\ $$ Answered by mr W last updated on 01/Dec/25 $$\frac{\mathrm{1}}{{n}\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}} }=\frac{{A}}{{n}}+\frac{{B}}{\mathrm{2}{n}+\mathrm{1}}+\frac{{C}}{\left(\mathrm{2}{n}+\mathrm{1}\right)^{\mathrm{2}}…
Question Number 226515 by mr W last updated on 01/Dec/25 Commented by Ghisom_ last updated on 02/Dec/25 $${S}_{{k}} =\underset{{n}=\mathrm{1}} {\overset{{k}} {\sum}}\:\frac{{n}}{{n}^{\mathrm{4}} +{n}^{\mathrm{2}} +\mathrm{1}}\:=\frac{{k}\left({k}+\mathrm{1}\right)}{\mathrm{2}\left({k}^{\mathrm{2}} +{k}+\mathrm{1}\right)} \\…
Question Number 226471 by Rojarani last updated on 30/Nov/25 $$\:{If},\:{x}^{\mathrm{2}} +\mathrm{2}{y}^{\mathrm{2}} \infty{xy}\: \\ $$$$\:\:{then}\:{prove}\:{that},\:\mathrm{2}{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \infty{xy} \\ $$ Commented by fantastic2 last updated on 30/Nov/25…
Question Number 226409 by hardmath last updated on 27/Nov/25 Answered by fantastic2 last updated on 27/Nov/25 Commented by fantastic2 last updated on 27/Nov/25 $$\alpha+\mathrm{90}^{\mathrm{0}} +\mathrm{70}^{\mathrm{0}}…
Question Number 226362 by Rojarani last updated on 26/Nov/25 Answered by Frix last updated on 26/Nov/25 $$\alpha=\sqrt{\mathrm{3}}\mathrm{e}^{−\mathrm{i}\frac{\mathrm{3}\pi}{\mathrm{4}}} \\ $$$$\beta=\sqrt{\mathrm{3}}\mathrm{e}^{\mathrm{i}\frac{\mathrm{3}\pi}{\mathrm{4}}} \\ $$$$\alpha^{{n}} +\beta^{{n}} =\mathrm{2}×\mathrm{3}^{\frac{{n}}{\mathrm{2}}} \mathrm{cos}\:\frac{\mathrm{3}{n}\pi}{\mathrm{4}} \\…
Question Number 226176 by Spillover last updated on 22/Nov/25 Commented by fantastic2 last updated on 22/Nov/25 $${you}\:{forgot}\:{dx} \\ $$ Answered by Spillover last updated on…
Question Number 226117 by Spillover last updated on 20/Nov/25 Answered by Spillover last updated on 20/Nov/25 Answered by Spillover last updated on 20/Nov/25 Answered by…
Question Number 226149 by Spillover last updated on 20/Nov/25 Answered by Spillover last updated on 21/Nov/25 Answered by Spillover last updated on 21/Nov/25 Terms of…
Question Number 226144 by Linton last updated on 20/Nov/25 $$\mathrm{6}^{{x}} \:+\:\mathrm{6}^{{y}} \:=\:\mathrm{42} \\ $$$${x}\:+\:{y}\:=\mathrm{3} \\ $$$${Find}\:{x}\:{and}\:{y}\:? \\ $$ Commented by mr W last updated on…
Question Number 226118 by Spillover last updated on 20/Nov/25 Answered by mr W last updated on 20/Nov/25 $$\mathrm{1}+\alpha+\beta+\gamma+\alpha\beta+\beta\gamma+\gamma\alpha+\alpha\beta\gamma=\mathrm{33} \\ $$$$\mathrm{1}+\mathrm{6}+\alpha\beta+\beta\gamma+\gamma\alpha+\alpha\beta\gamma=\mathrm{33} \\ $$$$\alpha\beta+\beta\gamma+\gamma\alpha+\alpha\beta\gamma=\mathrm{26}\:\:\:…\left({i}\right) \\ $$$$\left(\alpha+\beta+\gamma\right)^{\mathrm{3}} =\alpha^{\mathrm{3}}…