Expand $\frac{(1-2x)^{\frac{1}{3}}}{1+x}$ up to and including the term in $x^2$. State the commo

Question

Expand $\frac{(1-2x)^{\frac{1}{3}}}{1+x}$ up to and including the term in $x^2$. State the common open range of validity. Explain why substituting $x=\frac34$ is not justified by these expansions.[7]

Accepted Answer

$1-\frac53x+\frac{11}9x^2+\cdots;\ |x|<\frac12$. The substitution is outside this range.

Combining two validity conditions

Combining two validity conditions

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