What is taylor e^{-3x},0 ?

The answer to taylor e^{-3x},0 is 1-3x+9/2 x^2-9/2 x^3+27/8 x^4+\ldots

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taylor e^{-3x},0

taylor

e^{- 3 x}, 0

1 - 3 x + \frac{9}{2} x^{2} - \frac{9}{2} x^{3} + \frac{27}{8} x^{4} + \dots

Solution steps

Apply the Taylor Formula

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

Evaluate Derivatives

= 1 + \frac{- 3}{1 !} x + \frac{9}{2 !} x^{2} + \frac{- 27}{3 !} x^{3} + \frac{81}{4 !} x^{4} + \dots

Refine

= 1 - 3 x + \frac{9}{2} x^{2} - \frac{9}{2} x^{3} + \frac{27}{8} x^{4} + \dots

n = 2 \sum \infty x^{n}

\frac{d y}{d x} = - \frac{sin ( x )}{y}

\int cos (r^{2}) d r

x y^{^{'}} = (1 - x) y + x^{2} + x^{3}

y^{^{''}} + 4 y^{^{'}} = cos^{2} (x)

What is taylor e^{-3x},0 ?
The answer to taylor e^{-3x},0 is 1-3x+9/2 x^2-9/2 x^3+27/8 x^4+\ldots