Taylor series of eˣ up to degree 8
The Taylor polynomial of degree 8 for eˣ about 0 (the Maclaurin polynomial) is 1 + x + x²/2 + x³/6 + x⁴/24 + x⁵/120 + x⁶/720 + x⁷/5040 + x⁸/40320. The full series converges for every x.
How to work it out
- 1
The Taylor series of f about 0 uses the derivatives of f at 0:
- 2
For f(x) = eˣ the first derivatives at 0 are:
- 3
Divide each by k! and keep the terms up to degree 8:
- 4
Euler's formula comes from putting x = iθ into this series: the even powers give cos θ, the odd powers give i sin θ.
- 5
Radius of convergence: ∞ (it converges for every x). Inside it, more terms give a better approximation; outside, the series does not settle.
LaTeX source of the formulas on this page
\[ e^{x} \approx 1 + x + \frac{1}{2}x^{2} + \frac{1}{6}x^{3} + \frac{1}{24}x^{4} + \frac{1}{120}x^{5} + \frac{1}{720}x^{6} + \frac{1}{5040}x^{7} + \frac{1}{40320}x^{8} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 1,\quad f^{(1)}(0) = 1,\quad f^{(2)}(0) = 1,\quad f^{(3)}(0) = 1,\quad f^{(4)}(0) = 1,\quad f^{(5)}(0) = 1,\quad f^{(6)}(0) = 1 \]
\[ e^{i\theta} = \cos\theta + i\sin\theta \]Degree 8 polynomial against the real eˣ
| x | Polynomial | Exact | Error |
|---|---|---|---|
| 0.1 | 1.1051709 | 1.1051709 | 3.11e-15 |
| 0.5 | 1.6487213 | 1.6487213 | 5.66e-9 |
| 0.9 | 2.4596019 | 2.4596031 | 1.17e-6 |
| 2 | 7.3873016 | 7.3890561 | 1.75e-3 |
Questions
What is the Taylor series of eˣ?
1 + x + x²/2 + x³/6 + x⁴/24 + x⁵/120 + x⁶/720 + x⁷/5040 + x⁸/40320 + …
What is the degree 8 Taylor polynomial of eˣ?
1 + x + x²/2 + x³/6 + x⁴/24 + x⁵/120 + x⁶/720 + x⁷/5040 + x⁸/40320
What is the radius of convergence of the series of eˣ?
∞ (it converges for every x)
What is the Maclaurin series?
A Taylor series centred at 0.