Relation between Weierstrass $\wp$-functions 
Let $\Lambda=[\lambda_1,\lambda_2]$ be a lattice with associated Weierstrass function $\wp$, and consider the Weierstrass function $\wp_2$ associated to the lattice $\Lambda_2=[\tfrac{1}{2}\lambda_1,\lambda_2]$. Prove the identities $$\wp_2(z)=\wp(z)+\wp(z+\tfrac{1}{2}\lambda_1)-\wp(\tfrac{1}{2}\lambda_1)$$

Recall that $\wp(z)=\dfrac{1}{z^2}+\sum_{0\neq \omega \in \Lambda}\left( \dfrac{1}{(z-\omega)^2}-\dfrac{1}{\omega^2}\right)$
Every $\omega \in \Lambda$ has the form $\omega=n_\omega \lambda_1 + m_\omega \lambda_2$. Let $\omega_2 \in \Lambda_2$ we have $\omega_2=\omega-\tfrac{1}{2}n_\omega\lambda_1$. Therefore
$$\wp_2(z)=\dfrac{1}{z^2}+\sum_{0\neq \omega \in \Lambda}\left( \dfrac{1}{(z-\omega+\tfrac{1}{2}n_\omega\lambda_1)^2}-\dfrac{1}{(\omega-\tfrac{1}{2}n_\omega\lambda_1)^2}\right)$$
$$\wp(\tfrac{1}{2}\lambda_1)=\dfrac{4}{\lambda_1^2}+\sum_{0 \neq \omega \in \Lambda}\left(\dfrac{1}{(\tfrac{1}{2}\lambda_1-\omega)^2}-\dfrac{1}{\omega^2}\right)$$
$$\wp(z+\tfrac{1}{2}\lambda_1)=\dfrac{1}{(z+\tfrac{1}{2}\lambda_1)^2}+\sum_{0 \neq \omega \in \Lambda} \left( \dfrac{1}{(z-\omega+\tfrac{1}{2}\lambda_1)^2}- \dfrac{1}{\omega^2}\right)$$
I'm stuck here because I don't know how to deal with these infinite sum.

Given a lattice $\,\Lambda,\,$ the Weierstrass $\wp$ function is characterized by being a
meromorphic doubly periodic function with period
lattice $\,\Lambda\,$ whose only poles are at points in $\,\Lambda,\,$
 and whose Laurent series at the origin is
$\,\wp (z) = z^{-2} + O(z^2)$. In your first equation, note
that the right side satisfies the poles and Laurent series
properties for the lattice $\,\Lambda_2.$
The Wikipedia article Weierstrass elliptic functions states

Further development of the theory of elliptic functions shows that Weierstrass's function is determined up to addition of a constant and multiplication by a non-zero constant by the position and type of the poles alone, amongst all meromorphic functions with the given period lattice.

