= Termwise derivative proof of Weierstrass periodicity
{title2=$\wp'(z)=-2\sum_{\omega\in\Lambda}(z-\omega)^{-3}$}
The <Normal convergence of the Weierstrass elliptic-function series> allows its <derivative> to be written as an absolutely locally uniformly convergent sum of inverse cubes. For a <period lattice> $\Lambda$, this derivative sum is invariant under translation by any lattice vector, by reindexing. Consequently $\wp(z+\omega)-\wp(z)$ is constant; its apparent <poles> cancel. The <Weierstrass elliptic function> is even, so for a primitive lattice generator $\omega$, evaluation at $z=-\omega/2$ makes that constant zero. Applying this to two generators proves that this is a <periodic function>, namely $\wp$. This argument avoids separating the original convergent series into two divergent inverse-square sums.
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