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Calculus · Taylor-Series

Taylor series of atan(x)

\operatorname{atan}{\left(x \right)}

Step by step

  1. f(x) = \operatorname{atan}{\left(x \right)}

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

  2. f^{(0)}(0) = \operatorname{atan}{\left(x \right)}\big|_{x=0} = 0

    Derivative 0 at the centre.

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

    Derivative 1 at the centre.

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

    Derivative 2 at the centre.

  5. f^{(3)}(0) = \frac{2 \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right)}{\left(x^{2} + 1\right)^{2}}\big|_{x=0} = -2

    Derivative 3 at the centre.

  6. f^{(4)}(0) = \frac{24 x \left(- \frac{2 x^{2}}{x^{2} + 1} + 1\right)}{\left(x^{2} + 1\right)^{3}}\big|_{x=0} = 0

    Derivative 4 at the centre.

  7. f^{(5)}(0) = \frac{24 \left(\frac{16 x^{4}}{\left(x^{2} + 1\right)^{2}} - \frac{12 x^{2}}{x^{2} + 1} + 1\right)}{\left(x^{2} + 1\right)^{3}}\big|_{x=0} = 24

    Derivative 5 at the centre.

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

    Assemble the terms up to degree 5.

Reveal the answer
\frac{x^{5}}{5} - \frac{x^{3}}{3} + x + O((x-0)^{6})