Taylor series of arcsin x up to degree 5
The Taylor polynomial of degree 5 for arcsin x about 0 (the Maclaurin polynomial) is x + x³/6 + 3x⁵/40. 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) = arcsin 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
\[ \arcsin\left(x\right) \approx x + \frac{1}{6}x^{3} + \frac{3}{40}x^{5} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 0,\quad f^{(1)}(0) = 1,\quad f^{(2)}(0) = 0,\quad f^{(3)}(0) = 1,\quad f^{(4)}(0) = 0,\quad f^{(5)}(0) = 9 \]Degree 5 polynomial against the real arcsin x
| x | Polynomial | Exact | Error |
|---|---|---|---|
| 0.1 | 0.10016742 | 0.10016742 | 4.49e-9 |
| 0.5 | 0.52317708 | 0.52359878 | 4.22e-4 |
| 0.9 | 1.0657868 | 1.1197695 | 5.40e-2 |
| 2 | 5.7333333 | undefined | - |
Questions
What is the Taylor series of arcsin x?
x + x³/6 + 3x⁵/40 + 5x⁷/112 + …
What is the degree 5 Taylor polynomial of arcsin x?
x + x³/6 + 3x⁵/40
What is the radius of convergence of the series of arcsin x?
1
What is the Maclaurin series?
A Taylor series centred at 0.