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Calculus · Taylor-Series

Taylor series of e^(-x^2)

e^{- x^{2}}

Step by step

  1. f(x) = e^{- x^{2}}

    Taylor series about x = 0: f(x) = Σ fᵏ(x₀)/k! · (x − x₀)ᵏ.

  2. f^{(0)}(0) = e^{- x^{2}}\big|_{x=0} = 1

    Derivative 0 at the centre.

  3. f^{(1)}(0) = - 2 x e^{- x^{2}}\big|_{x=0} = 0

    Derivative 1 at the centre.

  4. f^{(2)}(0) = 2 \left(2 x^{2} - 1\right) e^{- x^{2}}\big|_{x=0} = -2

    Derivative 2 at the centre.

  5. f^{(3)}(0) = 4 x \left(3 - 2 x^{2}\right) e^{- x^{2}}\big|_{x=0} = 0

    Derivative 3 at the centre.

  6. f^{(4)}(0) = 4 \left(4 x^{4} - 12 x^{2} + 3\right) e^{- x^{2}}\big|_{x=0} = 12

    Derivative 4 at the centre.

  7. f^{(5)}(0) = 8 x \left(- 4 x^{4} + 20 x^{2} - 15\right) e^{- x^{2}}\big|_{x=0} = 0

    Derivative 5 at the centre.

  8. \frac{x^{4}}{2} - x^{2} + 1

    Assemble the terms up to degree 5.

Reveal the answer
\frac{x^{4}}{2} - x^{2} + 1 + O((x-0)^{6})