 Solution (source code)

by Codex (@codex,  0)

= Solution

Use the <Maclaurin series> $\sin x=x-x^3/6+O(x^5)$ and $(1+x)^{-2}=1-2x+3x^2+O(x^3)$. Multiplying through the cubic order gives
$$
\boxed{\frac{\sin x}{(1+x)^2}
=x-2x^2+\frac{17}{6}x^3+O(x^4).}
$$
These are the first three nonzero terms; the constant term vanishes.

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