Taylor series of ā(1 + x) up to degree 3
The Taylor polynomial of degree 3 for ā(1 + x) about 0 (the Maclaurin polynomial) is 1 + x/2 - xĀ²/8 + xĀ³/16. 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) = ā(1 + x) the first derivatives at 0 are:
- 3
Divide each by k! and keep the terms up to degree 3:
- 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
\[ \sqrt{1 + x} \approx 1 + \frac{1}{2}x - \frac{1}{8}x^{2} + \frac{1}{16}x^{3} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 1,\quad f^{(1)}(0) = \frac{1}{2},\quad f^{(2)}(0) = -\frac{1}{4},\quad f^{(3)}(0) = \frac{3}{8} \]Degree 3 polynomial against the real ā(1 + x)
| x | Polynomial | Exact | Error |
|---|---|---|---|
| 0.1 | 1.0488125 | 1.0488088 | 3.65e-6 |
| 0.5 | 1.2265625 | 1.2247449 | 1.82e-3 |
| 0.9 | 1.3943125 | 1.3784049 | 1.59e-2 |
| 2 | 2.0000000 | 1.7320508 | 2.68e-1 |
Questions
What is the Taylor series of ā(1 + x)?
1 + x/2 - xĀ²/8 + xĀ³/16 - 5xā“/128 + 7xāµ/256 - 21xā¶/1024 + 33xā·/2048 - 429xāø/32768 + ā¦
What is the degree 3 Taylor polynomial of ā(1 + x)?
1 + x/2 - xĀ²/8 + xĀ³/16
What is the radius of convergence of the series of ā(1 + x)?
1
What is the Maclaurin series?
A Taylor series centred at 0.