Taylor series of arctan x up to degree 9
The Taylor polynomial of degree 9 for arctan x about 0 (the Maclaurin polynomial) is x - x³/3 + x⁵/5 - x⁷/7 + x⁹/9. The full series converges for |x| < 1.
How to work it out
- 1
The Taylor series of f about 0 uses the derivatives of f at 0:
- 2
For f(x) = arctan x the first derivatives at 0 are:
- 3
Divide each by k! and keep the terms up to degree 9:
- 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
\[ \arctan\left(x\right) \approx x - \frac{1}{3}x^{3} + \frac{1}{5}x^{5} - \frac{1}{7}x^{7} + \frac{1}{9}x^{9} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 0,\quad f^{(1)}(0) = 1,\quad f^{(2)}(0) = 0,\quad f^{(3)}(0) = -2,\quad f^{(4)}(0) = 0,\quad f^{(5)}(0) = 24,\quad f^{(6)}(0) = 0 \]Degree 9 polynomial against the real arctan x
| x | Polynomial | Exact | Error |
|---|---|---|---|
| 0.1 | 0.099668652 | 0.099668652 | 9.01e-13 |
| 0.5 | 0.46368428 | 0.46364761 | 3.67e-5 |
| 0.9 | 0.74981659 | 0.73281510 | 1.70e-2 |
| 2 | 44.336508 | 1.1071487 | 4.32e+1 |
Questions
What is the Taylor series of arctan x?
x - x³/3 + x⁵/5 - x⁷/7 + …
What is the degree 9 Taylor polynomial of arctan x?
x - x³/3 + x⁵/5 - x⁷/7 + x⁹/9
What is the radius of convergence of the series of arctan x?
1
What is the Maclaurin series?
A Taylor series centred at 0.