In the Arithmetic of Elliptic curves, VI.2 Theorem 3.1 (a), Silverman states the Eisenstein series $G_{2k}(\Lambda) = \sum_{\omega \in \Lambda \backslash 0} \frac{1}{\omega^{2k}}$ converges absolutely for all $k > 1$. In Theorem 3.1(b) he says part (a) implies that the Weierstrass $\wp$ function is also absolutely convergent. But in the proof of this he only bounds each term of the Weierstrass function by the following: $$ \left | \frac{1}{(z-\omega)^2} - \frac{1}{\omega^2} \right | \leq \frac{10 |z| }{ |\omega|^2}$$ (where $\omega$ is the lattice point). Isn't this insufficient because $G_{2k}(\Lambda)$ does not converge for $k=1$? I see other sources (e.g. https://math.berkeley.edu/~dcorwin/math185S20/SS-elliptic.pdf) bound each term by $O(\omega^{-3})$, and that argument I do follow.
Am I missing something here in Silverman's proof?
This mistake is noted on p. 18 of the errata on Silverman's webpage.