Taylor series of cosh x up to degree 10
The Taylor polynomial of degree 10 for cosh x about 0 (the Maclaurin polynomial) is 1 + x²/2 + x⁴/24 + x⁶/720 + x⁸/40320 + x¹⁰/3628800. 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) = cosh x the first derivatives at 0 are:
- 3
Divide each by k! and keep the terms up to degree 10:
- 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
\[ \cosh\left(x\right) \approx 1 + \frac{1}{2}x^{2} + \frac{1}{24}x^{4} + \frac{1}{720}x^{6} + \frac{1}{40320}x^{8} + \frac{1}{3628800}x^{10} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 1,\quad f^{(1)}(0) = 0,\quad f^{(2)}(0) = 1,\quad f^{(3)}(0) = 0,\quad f^{(4)}(0) = 1,\quad f^{(5)}(0) = 0,\quad f^{(6)}(0) = 1 \]Degree 10 polynomial against the real cosh x
| x | Polynomial | Exact | Error |
|---|---|---|---|
| 0.1 | 1.0050042 | 1.0050042 | 2.22e-16 |
| 0.5 | 1.1276260 | 1.1276260 | 5.10e-13 |
| 0.9 | 1.4330864 | 1.4330864 | 5.92e-10 |
| 2 | 3.7621869 | 3.7621957 | 8.74e-6 |
Questions
What is the Taylor series of cosh x?
1 + x²/2 + x⁴/24 + x⁶/720 + x⁸/40320 + …
What is the degree 10 Taylor polynomial of cosh x?
1 + x²/2 + x⁴/24 + x⁶/720 + x⁸/40320 + x¹⁰/3628800
What is the radius of convergence of the series of cosh x?
∞
What is the Maclaurin series?
A Taylor series centred at 0.