Question Number 201209 by sonukgindia last updated on 02/Dec/23 Commented by ajfour last updated on 02/Dec/23 should it converge ? Commented by mr W last updated on 02/Dec/23…
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Question Number 201070 by Rodier97 last updated on 29/Nov/23 $$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{Un}\:=\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{\mathrm{1}}{\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}} \\ $$$$ \\ $$$${show}\:\:{that}\:{the}\:{sequence}\:{converges}\:{and} \\ $$$${determine}\:{the}\:{limit}\: \\ $$$$ \\…
Question Number 201091 by MrGHK last updated on 29/Nov/23 Commented by Frix last updated on 29/Nov/23 $$\mathrm{Look}\:\mathrm{at}\:\mathrm{the}\:\mathrm{first}\:\mathrm{few}\:\mathrm{summands}: \\ $$$${i}=\mathrm{0}\:\rightarrow\:\mathrm{1} \\ $$$${i}=\mathrm{1}\:\rightarrow\:\frac{{n}}{\mathrm{2}{n}+\mathrm{4}} \\ $$$${i}=\mathrm{2}\:\rightarrow\:\frac{{n}^{\mathrm{2}} −{n}}{\mathrm{6}{n}+\mathrm{24}{n}+\mathrm{24}} \\…
Question Number 201035 by sonukgindia last updated on 28/Nov/23 Commented by mr W last updated on 28/Nov/23 $${the}\:{side}\:{length}\:{of}\:{the}\:{square}\:{can}\:{not} \\ $$$${be}\:{equal}\:{to}\:{the}\:{radius}\:{of}\:{the}\:{smaller} \\ $$$${semi}−{circles}! \\ $$ Commented…
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Question Number 201027 by sonukgindia last updated on 28/Nov/23 Answered by Atomist last updated on 28/Nov/23 $${sechx}=\frac{\mathrm{1}}{{coshx}}\: \\ $$$${coshx}=\frac{{e}^{{x}} +{e}^{−{x}} }{\mathrm{2}} \\ $$$$\int{sechxdx}=\int\frac{\mathrm{2}}{{e}^{{x}} +{e}^{−{x}} }{dx}=…
Question Number 200984 by sonukgindia last updated on 27/Nov/23 Answered by Calculusboy last updated on 29/Nov/23 $$\mathrm{2} \\ $$ Answered by witcher3 last updated on…
Question Number 200980 by sonukgindia last updated on 27/Nov/23 Answered by MM42 last updated on 27/Nov/23 $$\sqrt{{x}+\sqrt{{x}+\sqrt{{x}…}}}={y}\Rightarrow\:{x}={y}^{\mathrm{2}} −{y}\Rightarrow{dx}=\left(\mathrm{2}{y}−\mathrm{1}\right){dy} \\ $$$${y}=\frac{\mathrm{1}+\sqrt{\mathrm{1}+\mathrm{4}{x}}}{\mathrm{2}}\:\:\:\:\:\Rightarrow\:\:\:{x}=\mathrm{0}\rightarrow{y}=\mathrm{1}\:\:\:\:;\:\:{x}=\mathrm{1}\rightarrow{y}=\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$$\left.\Rightarrow\int_{\mathrm{1}} ^{\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}} \:\:\frac{\mathrm{2}{y}−\mathrm{1}}{{y}}\:{dy}\:=\left(\mathrm{2}{y}−{lny}\right)\right]_{\mathrm{1}} ^{\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}}…