Solution
= Solution
Set $h=x+1$, the displacement from the expansion point. The numerator has <Taylor series> $\sinh h=h+h^3/6+O(h^5)$, while the reciprocal denominator has <geometric series> $(1+h)^{-1}=1-h+h^2-h^3+h^4+O(h^5)$. Multiplication gives
$$
\boxed{\frac{\sinh(x+1)}{x+2}=(x+1)-(x+1)^2+\frac76(x+1)^3-\frac76(x+1)^4+O((x+1)^5).}
$$
The local <power series> converges for $|x+1|<1$, the distance to the denominator's zero at $x=-2$.