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Calculus · Taylor-Series

Taylor series of log(x + 1)

\log{\left(x + 1 \right)}

Шаг за шагом

  1. f(x) = \log{\left(x + 1 \right)}

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

  2. f^{(0)}(0) = \log{\left(x + 1 \right)}\big|_{x=0} = 0

    Derivative 0 at the centre.

  3. f^{(1)}(0) = \frac{1}{x + 1}\big|_{x=0} = 1

    Derivative 1 at the centre.

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

    Derivative 2 at the centre.

  5. f^{(3)}(0) = \frac{2}{\left(x + 1\right)^{3}}\big|_{x=0} = 2

    Derivative 3 at the centre.

  6. f^{(4)}(0) = - \frac{6}{\left(x + 1\right)^{4}}\big|_{x=0} = -6

    Derivative 4 at the centre.

  7. f^{(5)}(0) = \frac{24}{\left(x + 1\right)^{5}}\big|_{x=0} = 24

    Derivative 5 at the centre.

  8. \frac{x^{5}}{5} - \frac{x^{4}}{4} + \frac{x^{3}}{3} - \frac{x^{2}}{2} + x

    Assemble the terms up to degree 5.

Откройте ответ.
\frac{x^{5}}{5} - \frac{x^{4}}{4} + \frac{x^{3}}{3} - \frac{x^{2}}{2} + x + O((x-0)^{6})