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Taylor series of tan x up to degree 3

The Taylor polynomial of degree 3 for tan x about 0 (the Maclaurin polynomial) is x + x³/3. The full series converges for |x| < π/2.

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tan x and its degree 3 Taylor polynomial-2-112-15-10-551015tan xdegree 3
tan x and its degree 3 Taylor polynomial. The polynomial hugs the curve inside the radius of convergence (π/2) 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) = tan x the first derivatives at 0 are:

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

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

    tan⁡(x)≈x+13x3\tan\left(x\right) \approx x + \frac{1}{3}x^{3}
  4. 4

    Radius of convergence: π/2 ≈ 1.5708. Inside it, more terms give a better approximation; outside, the series does not settle.

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

Degree 3 polynomial against the real tan x

xPolynomialExactError
0.10.100333330.100334671.34e-6
0.50.541666670.546302494.64e-3
0.91.14300001.26015821.17e-1
24.6666667-2.18503996.85e+0

Questions

What is the Taylor series of tan x?

x + x³/3 + 2x⁵/15 + 17x⁷/315 + …

What is the degree 3 Taylor polynomial of tan x?

x + x³/3

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

π/2 ≈ 1.5708

What is the Maclaurin series?

A Taylor series centred at 0.

Other degrees of tan x

Degree 3 of other functions

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