WordGPT

Taylor series of arctan x up to degree 7

The Taylor polynomial of degree 7 for arctan x about 0 (the Maclaurin polynomial) is x - x³/3 + x⁵/5 - x⁷/7. The full series converges for |x| < 1.

Explain it step by step in chatMake a short video of it
arctan x and its degree 7 Taylor polynomial-1.5-1-0.50.511.5-1-0.50.51arctan xdegree 7
arctan x and its degree 7 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)=−2,f(4)(0)=0,f(5)(0)=24,f(6)(0)=0f^{(0)}(0) = 0,\quad f^{(1)}(0) = 1,\quad f^{(2)}(0) = 0,\quad f^{(3)}(0) = -2,\quad f^{(4)}(0) = 0,\quad f^{(5)}(0) = 24,\quad f^{(6)}(0) = 0
  3. 3

    Divide each by k! and keep the terms up to degree 7:

    arctan⁡(x)≈x−13x3+15x5−17x7\arctan\left(x\right) \approx x - \frac{1}{3}x^{3} + \frac{1}{5}x^{5} - \frac{1}{7}x^{7}
  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
\[ \arctan\left(x\right) \approx x - \frac{1}{3}x^{3} + \frac{1}{5}x^{5} - \frac{1}{7}x^{7} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 0,\quad f^{(1)}(0) = 1,\quad f^{(2)}(0) = 0,\quad f^{(3)}(0) = -2,\quad f^{(4)}(0) = 0,\quad f^{(5)}(0) = 24,\quad f^{(6)}(0) = 0 \]

Degree 7 polynomial against the real arctan x

xPolynomialExactError
0.10.0996686520.0996686521.10e-10
0.50.463467260.463647611.80e-4
0.90.706769870.732815102.60e-2
2-12.5523811.10714871.37e+1

Questions

What is the Taylor series of arctan x?

x - x³/3 + x⁵/5 - x⁷/7 + …

What is the degree 7 Taylor polynomial of arctan x?

x - x³/3 + x⁵/5 - x⁷/7

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 7 of other functions

    Taylor series of arctan x: degree 7 polynomial and error | WordGPT