Taylor series of ln(1 + x) up to degree 5
The Taylor polynomial of degree 5 for ln(1 + x) about 0 (the Maclaurin polynomial) is x - x²/2 + x³/3 - x⁴/4 + 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) = ln(1 + 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 (converges for −1 < x ≤ 1). Inside it, more terms give a better approximation; outside, the series does not settle.
LaTeX source of the formulas on this page
\[ \ln\left(1 + x\right) \approx x - \frac{1}{2}x^{2} + \frac{1}{3}x^{3} - \frac{1}{4}x^{4} + \frac{1}{5}x^{5} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 0,\quad f^{(1)}(0) = 1,\quad f^{(2)}(0) = -1,\quad f^{(3)}(0) = 2,\quad f^{(4)}(0) = -6,\quad f^{(5)}(0) = 24 \]Degree 5 polynomial against the real ln(1 + x)
| x | Polynomial | Exact | Error |
|---|---|---|---|
| 0.1 | 0.095310333 | 0.095310180 | 1.54e-7 |
| 0.5 | 0.40729167 | 0.40546511 | 1.83e-3 |
| 0.9 | 0.69207300 | 0.64185389 | 5.02e-2 |
| 2 | 5.0666667 | 1.0986123 | 3.97e+0 |
Questions
What is the Taylor series of ln(1 + x)?
x - x²/2 + x³/3 - x⁴/4 + x⁵/5 - x⁶/6 + x⁷/7 - x⁸/8 + …
What is the degree 5 Taylor polynomial of ln(1 + x)?
x - x²/2 + x³/3 - x⁴/4 + x⁵/5
What is the radius of convergence of the series of ln(1 + x)?
1 (converges for −1 < x ≤ 1)
What is the Maclaurin series?
A Taylor series centred at 0.