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Taylor series of arctan x up to degree 3

The Taylor polynomial of degree 3 for arctan x about 0 (the Maclaurin polynomial) is x - x³/3. The full series converges for |x| < 1.

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arctan x and its degree 3 Taylor polynomial-1.5-1-0.50.511.5-1-0.50.51arctan xdegree 3
arctan x and its degree 3 Taylor polynomial. The polynomial hugs the curve inside the radius of convergence (1) and leaves it outside.

How to work it out

  1. 1

    The Taylor series of f about 0 uses the derivatives of f at 0:

    f(x)=∑k=0∞f(k)(0)k! xkf(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k
  2. 2

    For f(x) = arctan x the first derivatives at 0 are:

    f(0)(0)=0,f(1)(0)=1,f(2)(0)=0,f(3)(0)=−2f^{(0)}(0) = 0,\quad f^{(1)}(0) = 1,\quad f^{(2)}(0) = 0,\quad f^{(3)}(0) = -2
  3. 3

    Divide each by k! and keep the terms up to degree 3:

    arctan⁡(x)≈x−13x3\arctan\left(x\right) \approx x - \frac{1}{3}x^{3}
  4. 4

    Radius of convergence: 1. Inside it, more terms give a better approximation; outside, the series does not settle.

LaTeX source of the formulas on this page
\[ \arctan\left(x\right) \approx x - \frac{1}{3}x^{3} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 0,\quad f^{(1)}(0) = 1,\quad f^{(2)}(0) = 0,\quad f^{(3)}(0) = -2 \]

Degree 3 polynomial against the real arctan x

xPolynomialExactError
0.10.0996666670.0996686521.99e-6
0.50.458333330.463647615.31e-3
0.90.657000000.732815107.58e-2
2-0.666666671.10714871.77e+0

Questions

What is the Taylor series of arctan x?

x - x³/3 + x⁵/5 - x⁷/7 + …

What is the degree 3 Taylor polynomial of arctan x?

x - x³/3

What is the radius of convergence of the series of arctan x?

1

What is the Maclaurin series?

A Taylor series centred at 0.

Other degrees of arctan x

Degree 3 of other functions

    Taylor series of arctan x: degree 3 polynomial and error | WordGPT