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1-1-x-n-n-0-n-1-x-n-2-x-2-n-3-x-3-




Question Number 139286 by qaz last updated on 25/Apr/21
(1/((1−x)^n ))=( ((n),(0) ))+( ((n),(1) ))x+( ((n),(2) ))x^2 +( ((n),(3) ))x^3 +...
$$\frac{\mathrm{1}}{\left(\mathrm{1}−{x}\right)^{{n}} }=\left(\begin{pmatrix}{{n}}\\{\mathrm{0}}\end{pmatrix}\right)+\left(\begin{pmatrix}{{n}}\\{\mathrm{1}}\end{pmatrix}\right){x}+\left(\begin{pmatrix}{{n}}\\{\mathrm{2}}\end{pmatrix}\right){x}^{\mathrm{2}} +\left(\begin{pmatrix}{{n}}\\{\mathrm{3}}\end{pmatrix}\right){x}^{\mathrm{3}} +… \\ $$
Commented by mr W last updated on 25/Apr/21
wrong!  (1/((1−x)^n ))  = (((n−1)),(0) )+ ((n),(1) ) x+ (((n+1)),(2) ) x^2 + (((n+2)),(3) ) x^3 +...
$${wrong}! \\ $$$$\frac{\mathrm{1}}{\left(\mathrm{1}−{x}\right)^{{n}} } \\ $$$$=\begin{pmatrix}{{n}−\mathrm{1}}\\{\mathrm{0}}\end{pmatrix}+\begin{pmatrix}{{n}}\\{\mathrm{1}}\end{pmatrix}\:{x}+\begin{pmatrix}{{n}+\mathrm{1}}\\{\mathrm{2}}\end{pmatrix}\:{x}^{\mathrm{2}} +\begin{pmatrix}{{n}+\mathrm{2}}\\{\mathrm{3}}\end{pmatrix}\:{x}^{\mathrm{3}} +… \\ $$

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