For sufficiently small $x$, $(1+ax)^\alpha=1-3x+\frac{15}2x^2+\cdots$, where $a\ne0$ and

Question

For sufficiently small $x$, $(1+ax)^\alpha=1-3x+\frac{15}2x^2+\cdots$, where $a\ne0$ and $\alpha\in\mathbb Q$. Find $a$, $\alpha$, and the coefficient of $x^3$. State the open range of validity.[7]

Accepted Answer

$a=2,\ \alpha=-\frac32;\ -\frac{35}2;\ |x|<\frac12$

Recovering a rational exponent

Recovering a rational exponent

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