Taylor series of cos x up to degree 2
The Taylor polynomial of degree 2 for cos x about 0 (the Maclaurin polynomial) is 1 - x²/2. 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) = cos x the first derivatives at 0 are:
- 3
Divide each by k! and keep the terms up to degree 2:
- 4
Euler's formula: e^{iθ} = cos θ + i sin θ. The cos series collects the even powers of the series and the sin series the odd powers.
- 5
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
\[ \cos\left(x\right) \approx 1 - \frac{1}{2}x^{2} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 1,\quad f^{(1)}(0) = 0,\quad f^{(2)}(0) = -1 \]
\[ e^{i\theta} = \cos\theta + i\sin\theta \]Degree 2 polynomial against the real cos x
| x | Polynomial | Exact | Error |
|---|---|---|---|
| 0.1 | 0.99500000 | 0.99500417 | 4.17e-6 |
| 0.5 | 0.87500000 | 0.87758256 | 2.58e-3 |
| 0.9 | 0.59500000 | 0.62160997 | 2.66e-2 |
| 2 | -1.0000000 | -0.41614684 | 5.84e-1 |
Questions
What is the Taylor series of cos x?
1 - x²/2 + x⁴/24 - x⁶/720 + x⁸/40320 + …
What is the degree 2 Taylor polynomial of cos x?
1 - x²/2
What is the radius of convergence of the series of cos x?
∞
What is the Maclaurin series?
A Taylor series centred at 0.