Taylor series of arctan x up to degree 5
The Taylor polynomial of degree 5 for arctan x about 0 (the Maclaurin polynomial) is x - x³/3 + x⁵/5. 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 5:
- 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
\[ \arctan\left(x\right) \approx x - \frac{1}{3}x^{3} + \frac{1}{5}x^{5} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 0,\quad f^{(1)}(0) = 1,\quad f^{(2)}(0) = 0,\quad f^{(3)}(0) = -2,\quad f^{(4)}(0) = 0,\quad f^{(5)}(0) = 24 \]Degree 5 polynomial against the real arctan x
| x | Polynomial | Exact | Error |
|---|---|---|---|
| 0.1 | 0.099668667 | 0.099668652 | 1.42e-8 |
| 0.5 | 0.46458333 | 0.46364761 | 9.36e-4 |
| 0.9 | 0.77509800 | 0.73281510 | 4.23e-2 |
| 2 | 5.7333333 | 1.1071487 | 4.63e+0 |
Questions
What is the Taylor series of arctan x?
x - x³/3 + x⁵/5 - x⁷/7 + …
What is the degree 5 Taylor polynomial of arctan x?
x - x³/3 + x⁵/5
What is the radius of convergence of the series of arctan x?
1
What is the Maclaurin series?
A Taylor series centred at 0.