WordGPT

Taylor series of cos x up to degree 8

The Taylor polynomial of degree 8 for cos x about 0 (the Maclaurin polynomial) is 1 - x²/2 + x⁴/24 - x⁶/720 + x⁸/40320. The full series converges for every x.

Explain it step by step in chatMake a short video of it
cos x and its degree 8 Taylor polynomial-3-2-1123-1-0.50.51cos xdegree 8
cos x and its degree 8 Taylor polynomial. The polynomial follows the curve closely near 0 and drifts away further out; more terms stretch the good region.

How to work it out

  1. 1

    The Taylor series of f about 0 uses the derivatives of f at 0:

    f(x)=∑k=0∞f(k)(0)k! xkf(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k
  2. 2

    For f(x) = cos x the first derivatives at 0 are:

    f(0)(0)=1,f(1)(0)=0,f(2)(0)=−1,f(3)(0)=0,f(4)(0)=1,f(5)(0)=0,f(6)(0)=−1f^{(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
  3. 3

    Divide each by k! and keep the terms up to degree 8:

    cos⁡(x)≈1−12x2+124x4−1720x6+140320x8\cos\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}
  4. 4

    Euler's formula: e^{iθ} = cos θ + i sin θ. The cos series collects the even powers of the exe^{x} series and the sin series the odd powers.

    eiθ=cos⁡θ+isin⁡θe^{i\theta} = \cos\theta + i\sin\theta
  5. 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
\[ \cos\left(x\right) \approx 1 - \frac{1}{2}x^{2} + \frac{1}{24}x^{4} - \frac{1}{720}x^{6} + \frac{1}{40320}x^{8} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 1,\quad f^{(1)}(0) = 0,\quad f^{(2)}(0) = -1,\quad f^{(3)}(0) = 0,\quad f^{(4)}(0) = 1,\quad f^{(5)}(0) = 0,\quad f^{(6)}(0) = -1 \]
\[ e^{i\theta} = \cos\theta + i\sin\theta \]

Degree 8 polynomial against the real cos x

xPolynomialExactError
0.10.995004170.995004171.11e-16
0.50.877582560.877582562.69e-10
0.90.621610060.621609979.55e-8
2-0.41587302-0.416146842.74e-4

Questions

What is the Taylor series of cos x?

1 - x²/2 + x⁴/24 - x⁶/720 + x⁸/40320 + …

What is the degree 8 Taylor polynomial of cos x?

1 - x²/2 + x⁴/24 - x⁶/720 + x⁸/40320

What is the radius of convergence of the series of cos x?

∞

What is the Maclaurin series?

A Taylor series centred at 0.

Other degrees of cos x

Degree 8 of other functions

    Taylor series of cos x: degree 8 polynomial and error | WordGPT