Question Number 53467 by maxmathsup by imad last updated on 22/Jan/19
$${let}\:{A}_{{n}\:{m}} \:\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \left(\mathrm{1}−{x}\right)^{{m}} {dx}\:\:{with}\:{n}\:{and}\:{n}\:{integrs}\:{naturals} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}\:{m}} \:\:{by}\:{using}\:{factoriels} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\sum_{{n},{m}} \:{A}_{{nm}} \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 22/Jan/19
$$\left.\mathrm{1}\right){beta}\:{function}… \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {x}^{{n}+\mathrm{1}−\mathrm{1}} \left(\mathrm{1}−{x}\right)^{{m}+\mathrm{1}−\mathrm{1}} {dx}=\frac{\lceil\left({n}+\mathrm{1}\right)\lceil\left({m}+\mathrm{1}\right)}{\lceil\left({m}+{n}+\mathrm{2}\right)} \\ $$$${using}\:{beta}\:{function}… \\ $$$${answer}\:{is}=\frac{{n}!{m}!}{\left({m}+{n}+\mathrm{1}\right)!} \\ $$