Question Number 226743 by MrAjder last updated on 12/Dec/25 $$ \\ $$$${Prove}:\frac{\mathrm{1}}{\mathrm{2}{ne}}<\frac{\mathrm{1}}{{e}}−\left(\mathrm{1}−\frac{\mathrm{1}}{{n}}\right)^{{n}} <\frac{\mathrm{1}}{{ne}} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 226736 by ajfour last updated on 12/Dec/25 Commented by ajfour last updated on 12/Dec/25 $${Lets}\:{attempt}\:{the}\:\mathrm{25}\:{million}\:{dollar}\: \\ $$$${question}\:{that}\:{i}\:{solved}\:{then}\: \\ $$$${satisfactorily}\:{with} \\ $$$${sir}\:{MJS},\:{still}\:{in}\:{another}\:{better} \\ $$$${way}!…
Question Number 226738 by MrAjder last updated on 12/Dec/25 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\mathrm{Prove}}: \\ $$$$\underset{{v}\rightarrow\infty} {\mathrm{lim}}\sqrt{{n}}\left(\frac{\mathrm{1}}{\mathrm{2}}+{e}^{−{n}} \left(\underset{{k}=\mathrm{1}} {\overset{\mathrm{2}{n}} {\prod}}\begin{pmatrix}{\mathrm{2}{n}}\\{{k}}\end{pmatrix}^{\frac{\mathrm{1}}{\mathrm{2}{n}}} −\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{{n}^{{k}} }{{k}!}\right)\right)=\frac{\mathrm{1}}{\:\sqrt{\pi}}\left(\frac{{e}}{\mathrm{2}}−\frac{\sqrt{\mathrm{2}}}{\mathrm{3}}\right) \\ $$ Answered by TonyCWX…
Question Number 226719 by AgniMath last updated on 11/Dec/25 Commented by Ghisom_ last updated on 11/Dec/25 see there: https://math.stackexchange.com/questions/69519/closed-form-for-a-pair-of-continued-fractions Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 226728 by hardmath last updated on 11/Dec/25 $$\mathrm{Find}:\:\:\:\int_{\mathrm{0}} ^{\:\frac{\boldsymbol{\pi}}{\mathrm{4}}} \:\frac{\mathrm{dx}}{\mathrm{1}\:+\:\mathrm{sin}^{\mathrm{2}} \mathrm{x}}\:=\:? \\ $$ Answered by Ghisom_ last updated on 12/Dec/25 $$\mathrm{there}'\mathrm{s}\:\mathrm{an}\:\mathrm{easy}\:\mathrm{but}\:\mathrm{boring}\:\mathrm{solution}\:\mathrm{using} \\ $$$${t}=\mathrm{tan}\:{x}…
Question Number 226713 by ajfour last updated on 11/Dec/25 $$\:\:{If}\: \\ $$$${p}=\frac{\mathrm{1}}{{x}}−\frac{\mathrm{1}}{{x}^{\mathrm{2}} } \\ $$$${what}\:{should}\:{p}=\frac{\left({N}\right)_{\mathrm{4}} }{\left({D}\right)_{\mathrm{4}} } \\ $$$$\left({N}\right)_{\mathrm{4}} \:{means}\:{numerator}\:{of}\:{p}\:\:{in} \\ $$$${quaternary}\:{for}\:{x}\:{to}\:{be}\:\mathrm{777}? \\ $$ Terms…
Question Number 226714 by mr W last updated on 11/Dec/25 Commented by mr W last updated on 11/Dec/25 $${a}\:{dry}\:{rope}\:{with}\:{mass}\:{m}\:{and}\:{length}\:{l}\: \\ $$$${hangs}\:{free}\:{from}\:{two}\:{points}\:{with} \\ $$$${distance}\:{b}\:{to}\:{each}\:{other}.\:{for}\:{some} \\ $$$${reason}\:{the}\:{right}\:{half}\:{of}\:{the}\:{rope}\:…
Question Number 226721 by Spillover last updated on 11/Dec/25 Answered by breniam last updated on 12/Dec/25 $${x}^{\mathrm{2}} +{y}^{\mathrm{2}} +\mathrm{2}{xyy}'=\mathrm{0} \\ $$$${t}=\mathrm{ln}\mid{x}\mid \\ $$$${x}={e}^{{t}} \\ $$$${y}\left({x}\right)={z}\left({t}\right)…
Question Number 226702 by leromain last updated on 10/Dec/25 $$\int\frac{{ln}\left({x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{2}\right)}{{x}^{\mathrm{2}} +\mathrm{1}}{dx}\left[\right. \\ $$ Answered by Frix last updated on 11/Dec/25 $$\int\frac{\mathrm{ln}\:\left({x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{2}\right)}{{x}^{\mathrm{2}} +\mathrm{1}}{dx}=\int\frac{\mathrm{ln}\:\left(\left({x}+\mathrm{1}\right)\left({x}+\mathrm{2}\right)\right)}{{x}^{\mathrm{2}} +\mathrm{1}}{dx}=…
Question Number 226687 by fantastic2 last updated on 10/Dec/25 $${im}\:{back}\:{guys}! \\ $$$${my}\:{exams}\:{are}\:{over} \\ $$$$ \\ $$ Commented by fantastic2 last updated on 10/Dec/25 $${efgh} \\…