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

$\ln\left(\frac{1+x}{1-x}\right)=2x+\frac{2x^3}{3}+\frac{2x^5}{5}+\cdots$, valid for $-1<x<1$.

Series expansion involving natural logarithms

Series expansion involving natural logarithms

Solution:

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