Question Number 98116 by M±th+et+s last updated on 11/Jun/20
$${prove}\:{the}\:{compact}\:{form}\:{of}\:{multivariable} \\ $$$${Taylor}\:{series} \\ $$$$ \\ $$$$\left({T}\ast{B}\right)=\underset{{n}=\mathrm{1},{n}=\mathrm{2}..{n}_{{d}} =\mathrm{0}} {\overset{\infty} {\sum}}\frac{\prod_{\mathrm{i}=\mathrm{0}} ^{{n}_{{d}} } \left({x}_{\mathrm{i}} −{d}_{\mathrm{i}} \right)}{\prod_{\mathrm{i}=\mathrm{0}} ^{{n}_{{d}} } \left({n}_{\mathrm{i}} \right)!}\:\underset{\mathrm{i}=\mathrm{0}} {\overset{{n}_{{d}} } {\prod}}\frac{\partial}{\partial{x}_{\mathrm{i}} }{f} \\ $$