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Taylor series of arcsin x up to degree 3

The Taylor polynomial of degree 3 for arcsin x about 0 (the Maclaurin polynomial) is x + x³/6. The full series converges for |x| < 1.

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arcsin x and its degree 3 Taylor polynomial-1.5-1-0.50.511.5-1.5-1-0.50.511.5arcsin xdegree 3
arcsin x and its degree 3 Taylor polynomial. The polynomial hugs the curve inside the radius of convergence (1) and leaves it outside.

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) = arcsin x the first derivatives at 0 are:

    f(0)(0)=0,f(1)(0)=1,f(2)(0)=0,f(3)(0)=1f^{(0)}(0) = 0,\quad f^{(1)}(0) = 1,\quad f^{(2)}(0) = 0,\quad f^{(3)}(0) = 1
  3. 3

    Divide each by k! and keep the terms up to degree 3:

    arcsin⁡(x)≈x+16x3\arcsin\left(x\right) \approx x + \frac{1}{6}x^{3}
  4. 4

    Radius of convergence: 1. Inside it, more terms give a better approximation; outside, the series does not settle.

LaTeX source of the formulas on this page
\[ \arcsin\left(x\right) \approx x + \frac{1}{6}x^{3} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 0,\quad f^{(1)}(0) = 1,\quad f^{(2)}(0) = 0,\quad f^{(3)}(0) = 1 \]

Degree 3 polynomial against the real arcsin x

xPolynomialExactError
0.10.100166670.100167427.54e-7
0.50.520833330.523598782.77e-3
0.91.02150001.11976959.83e-2
23.3333333undefined-

Questions

What is the Taylor series of arcsin x?

x + x³/6 + 3x⁵/40 + 5x⁷/112 + …

What is the degree 3 Taylor polynomial of arcsin x?

x + x³/6

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

1

What is the Maclaurin series?

A Taylor series centred at 0.

Other degrees of arcsin x

Degree 3 of other functions

    Taylor series of arcsin x: degree 3 polynomial and error | WordGPT