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prove-the-compact-form-of-multivariable-Taylor-series-T-B-n-1-n-2-n-d-0-i-0-n-d-x-i-d-i-i-0-n-d-n-i-i-0-n-d-x-i-f-




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

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