Use the Maclaurin series of $\ln(1 + x)$ to show that \[ \ln \left(\frac{1 + x}{1 - x}\right) \app

Question

Use the Maclaurin series of $\ln(1 + x)$ to show that $$ \ln \left(\frac{1 + x}{1 - x}\right) \approx 2x + \frac{2}{3}x^3 + \frac{2}{5}x^5 + \dots $$ and state the range of $x$ for which the expansion is valid.

Accepted Answer

$|x| < 1.$

Series expansion involving natural logarithms

Series expansion involving natural logarithms