I'm working my way through Silverman and Tate's Undergraduate Introduction to Elliptic Curves. I haven't yet been able to study complex analysis, so it comes as no surprise that I'm having a tough time with that portion of the book right now.

Let $\omega_1, \omega_2 \in \mathbb{C}$ be two complex numbers which are $\mathbb{R}$-linearly independent and let: $$L = \mathbb{Z}\omega_1 + \mathbb{Z}\omega_2 = \{n_1\omega_1 + n_2\omega_2 : n_1, n_2 \in \mathbb{Z}\}$$ Let $\wp(u) = \frac{1}{u^2} + \sum\limits_{\omega \in L, \omega \neq 0} \left(\frac{1}{(u-\omega)^2} - \frac{1}{\omega^2}\right)$ Show that $\wp$ is a doubly periodic function, that is, show that $$\wp(u + \omega) = \wp(u)$$

If you are able, please give me a shove in the right direction. Thank you!

Dear Answerers: Thank you, I have been able to figure it out. Yes, convergence was quite tricky and I was trying to make this particular question much more difficult than it actually was. Thank you!