What is taylor x*e^{-x+1} ?

The answer to taylor x*e^{-x+1} is ex-ex^2+e/2 x^3-e/6 x^4+e/(24)x^5+\ldots

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taylor x*e^{-x+1}

taylor

x \cdot e^{- x + 1}

e x - e x^{2} + \frac{e}{2} x^{3} - \frac{e}{6} x^{4} + \frac{e}{24} x^{5} + \dots

Solution steps

Apply the Taylor Formula

= 0 + \frac{\frac{d}{d x} ( x e ^{- x + 1} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( x e ^{- x + 1} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( x e ^{- x + 1} ) ( 0 )}{3 !} x^{3} + \dots

Evaluate Derivatives

= 0 + \frac{e}{1 !} x + \frac{- 2 e}{2 !} x^{2} + \frac{3 e}{3 !} x^{3} + \frac{- 4 e}{4 !} x^{4} + \frac{5 e}{5 !} x^{5} + \dots

Refine

= e x - e x^{2} + \frac{e}{2} x^{3} - \frac{e}{6} x^{4} + \frac{e}{24} x^{5} + \dots

What is taylor x*e^{-x+1} ?
The answer to taylor x*e^{-x+1} is ex-ex^2+e/2 x^3-e/6 x^4+e/(24)x^5+\ldots