Taylor series of tan x up to degree 7
The Taylor polynomial of degree 7 for tan x about 0 (the Maclaurin polynomial) is x + x³/3 + 2x⁵/15 + 17x⁷/315. The full series converges for |x| < π/2.
How to work it out
- 1
The Taylor series of f about 0 uses the derivatives of f at 0:
- 2
For f(x) = tan x the first derivatives at 0 are:
- 3
Divide each by k! and keep the terms up to degree 7:
- 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
\[ \tan\left(x\right) \approx x + \frac{1}{3}x^{3} + \frac{2}{15}x^{5} + \frac{17}{315}x^{7} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 0,\quad f^{(1)}(0) = 1,\quad f^{(2)}(0) = 0,\quad f^{(3)}(0) = 2,\quad f^{(4)}(0) = 0,\quad f^{(5)}(0) = 16,\quad f^{(6)}(0) = 0 \]Degree 7 polynomial against the real tan x
| x | Polynomial | Exact | Error |
|---|---|---|---|
| 0.1 | 0.10033467 | 0.10033467 | 2.20e-11 |
| 0.5 | 0.54625496 | 0.54630249 | 4.75e-5 |
| 0.9 | 1.2475448 | 1.2601582 | 1.26e-2 |
| 2 | 15.841270 | -2.1850399 | 1.80e+1 |
Questions
What is the Taylor series of tan x?
x + x³/3 + 2x⁵/15 + 17x⁷/315 + …
What is the degree 7 Taylor polynomial of tan x?
x + x³/3 + 2x⁵/15 + 17x⁷/315
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.