WordGPT

Taylor series of ln(1 + x) up to degree 9

The Taylor polynomial of degree 9 for ln(1 + x) about 0 (the Maclaurin polynomial) is x - x²/2 + x³/3 - x⁴/4 + x⁵/5 - x⁶/6 + x⁷/7 - x⁸/8 + x⁹/9. The full series converges for |x| < 1.

Explain it step by step in chatMake a short video of it
ln(1 + x) and its degree 9 Taylor polynomial-1.5-1-0.50.511.5-2.5-2-1.5-1-0.50.51ln(1 + x)degree 9
ln(1 + x) and its degree 9 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) = ln(1 + x) the first derivatives at 0 are:

    f(0)(0)=0,f(1)(0)=1,f(2)(0)=−1,f(3)(0)=2,f(4)(0)=−6,f(5)(0)=24,f(6)(0)=−120f^{(0)}(0) = 0,\quad f^{(1)}(0) = 1,\quad f^{(2)}(0) = -1,\quad f^{(3)}(0) = 2,\quad f^{(4)}(0) = -6,\quad f^{(5)}(0) = 24,\quad f^{(6)}(0) = -120
  3. 3

    Divide each by k! and keep the terms up to degree 9:

    ln⁡(1+x)≈x−12x2+13x3−14x4+15x5−16x6+17x7−18x8+19x9\ln\left(1 + x\right) \approx x - \frac{1}{2}x^{2} + \frac{1}{3}x^{3} - \frac{1}{4}x^{4} + \frac{1}{5}x^{5} - \frac{1}{6}x^{6} + \frac{1}{7}x^{7} - \frac{1}{8}x^{8} + \frac{1}{9}x^{9}
  4. 4

    Radius of convergence: 1 (converges for −1 < x ≤ 1). Inside it, more terms give a better approximation; outside, the series does not settle.

LaTeX source of the formulas on this page
\[ \ln\left(1 + x\right) \approx x - \frac{1}{2}x^{2} + \frac{1}{3}x^{3} - \frac{1}{4}x^{4} + \frac{1}{5}x^{5} - \frac{1}{6}x^{6} + \frac{1}{7}x^{7} - \frac{1}{8}x^{8} + \frac{1}{9}x^{9} \]
\[ f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!}\,x^k \]
\[ f^{(0)}(0) = 0,\quad f^{(1)}(0) = 1,\quad f^{(2)}(0) = -1,\quad f^{(3)}(0) = 2,\quad f^{(4)}(0) = -6,\quad f^{(5)}(0) = 24,\quad f^{(6)}(0) = -120 \]

Degree 9 polynomial against the real ln(1 + x)

xPolynomialExactError
0.10.0953101800.0953101809.17e-12
0.50.405532300.405465116.72e-5
0.90.661065950.641853891.92e-2
237.5746031.09861233.65e+1

Questions

What is the Taylor series of ln(1 + x)?

x - x²/2 + x³/3 - x⁴/4 + x⁵/5 - x⁶/6 + x⁷/7 - x⁸/8 + …

What is the degree 9 Taylor polynomial of ln(1 + x)?

x - x²/2 + x³/3 - x⁴/4 + x⁵/5 - x⁶/6 + x⁷/7 - x⁸/8 + x⁹/9

What is the radius of convergence of the series of ln(1 + x)?

1 (converges for −1 < x ≤ 1)

What is the Maclaurin series?

A Taylor series centred at 0.

Other degrees of ln(1 + x)

Degree 9 of other functions

    Taylor series of ln(1 + x): degree 9 polynomial and error | WordGPT