Taylor series of e²ˣ up to degree 9
The Taylor polynomial of degree 9 for e²ˣ about 0 (the Maclaurin polynomial) is 1 + 2x + 2x² + 4x³/3 + 2x⁴/3 + 4x⁵/15 + 4x⁶/45 + 8x⁷/315 + 2x⁸/315 + 4x⁹/2835. 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) = e²ˣ the first derivatives at 0 are:
- 3
Divide each by k! and keep the terms up to degree 9:
- 4
Radius of convergence: ∞. Inside it, more terms give a better approximation; outside, the series does not settle.
LaTeX source of the formulas on this page
\[ e^{2 x} \approx 1 + 2x + 2x^{2} + \frac{4}{3}x^{3} + \frac{2}{3}x^{4} + \frac{4}{15}x^{5} + \frac{4}{45}x^{6} + \frac{8}{315}x^{7} + \frac{2}{315}x^{8} + \frac{4}{2835}x^{9} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 1,\quad f^{(1)}(0) = 2,\quad f^{(2)}(0) = 4,\quad f^{(3)}(0) = 8,\quad f^{(4)}(0) = 16,\quad f^{(5)}(0) = 32,\quad f^{(6)}(0) = 64 \]Degree 9 polynomial against the real e²ˣ
| x | Polynomial | Exact | Error |
|---|---|---|---|
| 0.1 | 1.2214028 | 1.2214028 | 2.89e-14 |
| 0.5 | 2.7182815 | 2.7182818 | 3.03e-7 |
| 0.9 | 6.0495302 | 6.0496475 | 1.17e-4 |
| 2 | 54.154145 | 54.598150 | 4.44e-1 |
Questions
What is the Taylor series of e²ˣ?
1 + 2x + 2x² + 4x³/3 + 2x⁴/3 + 4x⁵/15 + 4x⁶/45 + 8x⁷/315 + 2x⁸/315 + …
What is the degree 9 Taylor polynomial of e²ˣ?
1 + 2x + 2x² + 4x³/3 + 2x⁴/3 + 4x⁵/15 + 4x⁶/45 + 8x⁷/315 + 2x⁸/315 + 4x⁹/2835
What is the radius of convergence of the series of e²ˣ?
∞
What is the Maclaurin series?
A Taylor series centred at 0.