Question Number 200446 by hardmath last updated on 18/Nov/23 $${fund}\:\:\:\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\left(\mathrm{3}\:+\:\left(−\mathrm{1}\right)^{\boldsymbol{{n}}} \right)^{\boldsymbol{{n}}} }{{n}}\:{x}^{\boldsymbol{{n}}} \:=\:? \\ $$ Answered by mr W last updated on 19/Nov/23…
Question Number 200438 by hardmath last updated on 18/Nov/23 $$\mathrm{Find}: \\ $$$$\sqrt{\mathrm{3}\:+\:\sqrt{\mathrm{6}\:+\:\sqrt{\mathrm{9}\:+\:…\:+\:\sqrt{\mathrm{96}\:+\:\sqrt{\mathrm{99}}}}}}\:=\:? \\ $$ Commented by Frix last updated on 19/Nov/23 $$\mathrm{There}'\mathrm{s}\:\mathrm{nothing}\:\mathrm{to}\:\mathrm{find},\:\mathrm{just}\:\mathrm{use}\:\mathrm{a}\:\mathrm{good} \\ $$$$\mathrm{calculator}. \\…
Question Number 200432 by galivan last updated on 18/Nov/23 Commented by mr W last updated on 19/Nov/23 $${it}\:{seems}\:{you}\:{are}\:{using}\:{the}\:{forum}\:{just} \\ $$$${for}\:{editing}\:{and}\:{storing}\:{your}\:{own} \\ $$$${formulas}.\:{please}\:{use}\:{the}\:{editor} \\ $$$${function}\:{of}\:{this}\:{app}\:{for}\:{such}\:{purpose}! \\…
Question Number 200424 by hardmath last updated on 18/Nov/23 Commented by Frix last updated on 18/Nov/23 $$\mathrm{This}\:\mathrm{is}\:\mathrm{soft}\:\mathrm{math}… \\ $$$$\Omega=\frac{\pi\sqrt{\mathrm{2}}}{\mathrm{4}}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\mathrm{ln}\:\left(\mathrm{1}+\sqrt{\mathrm{2}}\right) \\ $$ Commented by hardmath last…
Question Number 200417 by hardmath last updated on 18/Nov/23 $$\mathrm{find}:\:\:\:\Omega\:=\:\int_{\mathrm{1}} ^{\:\infty} \:\frac{\sqrt{\mathrm{x}}}{\left(\mathrm{1}\:+\:\mathrm{x}^{\mathrm{2}} \right)}\:\mathrm{dx}\:=\:? \\ $$ Answered by witcher3 last updated on 18/Nov/23 $$\sqrt{\mathrm{x}}=\mathrm{y} \\ $$$$=\int_{\mathrm{1}}…
Question Number 200412 by hardmath last updated on 18/Nov/23 $$\mathrm{If} \\ $$$$\sqrt{\left(\mathrm{x}-\mathrm{6}\right)^{\mathrm{2}} \:+\:\left(\mathrm{y}+\mathrm{1}\right)^{\mathrm{2}} }\:+\:\sqrt{\left(\mathrm{x}-\mathrm{9}\right)^{\mathrm{2}} \:+\:\left(\mathrm{y}+\mathrm{5}\right)^{\mathrm{2}} } \\ $$$$ \\ $$$$\mathrm{find}:\:\:\:\mathrm{minumum}\:=\:? \\ $$ Answered by witcher3…
Question Number 200415 by Mingma last updated on 18/Nov/23 Answered by witcher3 last updated on 18/Nov/23 $$\Leftrightarrow\underset{\mathrm{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(−\frac{\mathrm{f}\left(\mathrm{x}\right)−\mathrm{f}\left(\mathrm{x}−\mathrm{h}\right)}{\mathrm{x}−\left(\mathrm{x}−\mathrm{h}\right)}\right)=−\mathrm{f}'\left(\mathrm{x}\right) \\ $$$$\Leftrightarrow−\mathrm{f}''\left(\mathrm{x}\right)=\underset{\mathrm{n}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{f}^{\left(\mathrm{n}\right)} \left(\mathrm{0}\right)\mathrm{x}^{\mathrm{n}} }{\mathrm{n}!} \\ $$$$=\mathrm{f}\left(\mathrm{x}\right)−\mathrm{f}\left(\mathrm{0}\right)=−\mathrm{f}''\left(\mathrm{x}\right)…
Question Number 200388 by mr W last updated on 18/Nov/23 $${find}\:{all}\:{values}\:{for}\:{k}\:{such}\:{that}\:{the}\:{eq}. \\ $$$${x}^{\mathrm{3}} −\mathrm{13}{x}+{k}=\mathrm{0}\:{has}\:{three}\:{integer}\:{roots}. \\ $$ Answered by Frix last updated on 18/Nov/23 $$\left({x}−{p}\right)\left({x}+\frac{{p}}{\mathrm{2}}−{q}\right)\left({x}+\frac{{p}}{\mathrm{2}}+{q}\right)=\mathrm{0} \\…
Question Number 200450 by hardmath last updated on 18/Nov/23 $${if}\:\:\:\frac{{sin}\mathrm{4}{x}}{{cos}\mathrm{6}{x}}\:=\:\mathrm{0}\:\:\:{find}\:\:\:{x}\:=\:? \\ $$ Answered by Frix last updated on 19/Nov/23 $$\mathrm{sin}\:\mathrm{4}{x}\:=\mathrm{0}\wedge\mathrm{cos}\:\mathrm{6}{x}\:\neq\mathrm{0}\:\Rightarrow\:{x}=\frac{{n}\pi}{\mathrm{2}} \\ $$ Terms of Service…
Question Number 200353 by faysal last updated on 17/Nov/23 $$\mathrm{If}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{x}^{\mathrm{3}} +\mathrm{3px}^{\mathrm{2}} +\mathrm{3qx}+\mathrm{r}=\mathrm{0}\:\mathrm{are} \\ $$$$\mathrm{in}\:\mathrm{harmonic}\:\mathrm{progression},\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\mathrm{2q}^{\mathrm{3}} =\mathrm{r}\left(\mathrm{3pq}−\mathrm{r}\right) \\ $$ Answered by jabarsing last updated on…