= Solution
There is a natural description
$$
H_c^*(X)\cong\varinjlim_{K\subseteq X\text{ compact}}H^*(X,X\setminus K).
$$
Closed balls form a cofinal family of compact subsets of $\mathbb R^n$. Excision and radial deformation identify every sufficiently large-ball term, and the resulting stable group is the local group at the origin:
$$
H_c^*(\mathbb R^n)\cong H^*(\mathbb R^n,\mathbb R^n\setminus\{0\}).
$$
It is $\mathbb Z$ in degree $n$ and zero otherwise.
Solved by gpt-5.6-sol high.
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