Taylor series of sinh x up to degree 9
The Taylor polynomial of degree 9 for sinh x about 0 (the Maclaurin polynomial) is x + x³/6 + x⁵/120 + x⁷/5040 + x⁹/362880. 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) = sinh x 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
\[ \sinh\left(x\right) \approx x + \frac{1}{6}x^{3} + \frac{1}{120}x^{5} + \frac{1}{5040}x^{7} + \frac{1}{362880}x^{9} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 0,\quad f^{(1)}(0) = 1,\quad f^{(2)}(0) = 0,\quad f^{(3)}(0) = 1,\quad f^{(4)}(0) = 0,\quad f^{(5)}(0) = 1,\quad f^{(6)}(0) = 0 \]Degree 9 polynomial against the real sinh x
| x | Polynomial | Exact | Error |
|---|---|---|---|
| 0.1 | 0.10016675 | 0.10016675 | 0.00e+0 |
| 0.5 | 0.52109531 | 0.52109531 | 1.23e-11 |
| 0.9 | 1.0265167 | 1.0265167 | 7.90e-9 |
| 2 | 3.6268078 | 3.6268604 | 5.26e-5 |
Questions
What is the Taylor series of sinh x?
x + x³/6 + x⁵/120 + x⁷/5040 + …
What is the degree 9 Taylor polynomial of sinh x?
x + x³/6 + x⁵/120 + x⁷/5040 + x⁹/362880
What is the radius of convergence of the series of sinh x?
∞
What is the Maclaurin series?
A Taylor series centred at 0.