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Taylor series of cosh x up to degree 4

The Taylor polynomial of degree 4 for cosh x about 0 (the Maclaurin polynomial) is 1 + x²/2 + x⁴/24. The full series converges for every x.

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cosh x and its degree 4 Taylor polynomial-3-2-112324681012cosh xdegree 4
cosh x and its degree 4 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) = cosh 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)=1f^{(0)}(0) = 1,\quad f^{(1)}(0) = 0,\quad f^{(2)}(0) = 1,\quad f^{(3)}(0) = 0,\quad f^{(4)}(0) = 1
  3. 3

    Divide each by k! and keep the terms up to degree 4:

    cosh⁡(x)≈1+12x2+124x4\cosh\left(x\right) \approx 1 + \frac{1}{2}x^{2} + \frac{1}{24}x^{4}
  4. 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
\[ \cosh\left(x\right) \approx 1 + \frac{1}{2}x^{2} + \frac{1}{24}x^{4} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 1,\quad f^{(1)}(0) = 0,\quad f^{(2)}(0) = 1,\quad f^{(3)}(0) = 0,\quad f^{(4)}(0) = 1 \]

Degree 4 polynomial against the real cosh x

xPolynomialExactError
0.11.00500421.00500421.39e-9
0.51.12760421.12762602.18e-5
0.91.43233751.43308647.49e-4
23.66666673.76219579.55e-2

Questions

What is the Taylor series of cosh x?

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

What is the degree 4 Taylor polynomial of cosh x?

1 + x²/2 + x⁴/24

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

∞

What is the Maclaurin series?

A Taylor series centred at 0.

Other degrees of cosh x

Degree 4 of other functions

    Taylor series of cosh x: degree 4 polynomial and error | WordGPT