Question Number 184349 by alcohol last updated on 05/Jan/23
$${I}_{{n}} \:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} {dx} \\ $$$${Relate}\:{I}_{{n}} \:{and}\:{I}_{{n}−\mathrm{1}} \\ $$$${Find}\:{I}_{{n}} \:{in}\:{terms}\:{of}\:{n} \\ $$$${hence}\:{deduce}\:{that}\:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} \begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}}{\mathrm{2}{k}+\mathrm{1}}=\frac{\mathrm{2}^{\mathrm{2}{n}} \left({n}!\right)^{\mathrm{2}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!} \\ $$
Answered by mr W last updated on 05/Jan/23
$${I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} {dx} \\ $$$$\:\:=\left[{x}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} \right]_{\mathrm{0}} ^{\mathrm{1}} +\mathrm{2}{n}\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{2}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}−\mathrm{1}} {dx} \\ $$$$\:\:=\mathrm{2}{n}\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}+{x}^{\mathrm{2}} −\mathrm{1}\right)\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}−\mathrm{1}} {dx} \\ $$$$\:\:=\mathrm{2}{n}\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}−\mathrm{1}} {dx}−\mathrm{2}{n}\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} {dx} \\ $$$$\:\:=\mathrm{2}{nI}_{{n}−\mathrm{1}} −\mathrm{2}{nI}_{{n}} \\ $$$$\Rightarrow{I}_{{n}} =\frac{\mathrm{2}{n}}{\mathrm{2}{n}+\mathrm{1}}{I}_{{n}−\mathrm{1}} \\ $$$${I}_{{n}} =\frac{\mathrm{2}{n}}{\mathrm{2}{n}+\mathrm{1}}×\frac{\mathrm{2}{n}−\mathrm{2}}{\mathrm{2}{n}−\mathrm{1}}×…×\frac{\mathrm{2}}{\mathrm{3}}×{I}_{\mathrm{0}} \\ $$$${I}_{\mathrm{0}} =\int_{\mathrm{0}} ^{\mathrm{1}} {dx}=\mathrm{1} \\ $$$$\Rightarrow{I}_{{n}} =\frac{\left(\mathrm{2}{n}\right)!!}{\left(\mathrm{2}{n}+\mathrm{1}\right)!!}=\frac{\left[\left(\mathrm{2}{n}\right)!!\right]^{\mathrm{2}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!}=\frac{\left[\mathrm{2}^{{n}} {n}!\right]^{\mathrm{2}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!}=\frac{\mathrm{2}^{\mathrm{2}{n}} \left({n}!\right)^{\mathrm{2}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!} \\ $$$$ \\ $$$${on}\:{the}\:{other}\:{side}: \\ $$$${I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} {dx} \\ $$$$\:\:=\int_{\mathrm{0}} ^{\mathrm{1}} \left[\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}\left(−{x}^{\mathrm{2}} \right)^{{k}} \right]{dx} \\ $$$$\:\:=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}} \begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}\left[\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{2}{k}} {dx}\right] \\ $$$$\:\:=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\mathrm{1}\right)^{{k}} \begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}\left[\frac{{x}^{\mathrm{2}{k}+\mathrm{1}} }{\mathrm{2}{k}+\mathrm{1}}\right]_{\mathrm{0}} ^{\mathrm{1}} \\ $$$$\:\:=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}}\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix} \\ $$$$\Rightarrow\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}}\begin{pmatrix}{{n}}\\{{k}}\end{pmatrix}\:=\frac{\mathrm{2}^{\mathrm{2}{n}} \left({n}!\right)^{\mathrm{2}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!} \\ $$