<s>[INST] Weierstrass Zeta function Series Let $\Gamma := \mathbb{Z} + \mathbb{Z} \tau$ be a lattice.
Then  \begin{align*} \zeta_\Gamma(z+1) - \zeta_\Gamma(z) \end{align*} is a function independent from $z$. Why is that so?
The function is defined as follows
\begin{align}
\zeta_\Gamma(z) = \frac{1}{z} + \sum_{(m,n)\neq (0,0)} \frac{1}{z - (m\tau+n)} + \frac{1}{m\tau+n} + \frac{z}{(m\tau+n)^2}
\end{align}
I see how the middle term of the sum cancels in this substraction but i dont really see how the $z$ vanish, especially the $\frac{1}{z+1} - \frac{1}{z}$ in front of the sum. 
\begin{align*}
\zeta_\Gamma(z+1)- \zeta_\Gamma(z+1) &= \frac{1}{z+1} + \sum_{(m,n+1)\neq (0,0)} \frac{1}{(z+1) - (m\tau+n)} + \frac{1}{m\tau+n} + \frac{(z+1)}{(m\tau+n)^2} \\ &-\left( \frac{1}{z} + \sum_{(m,n)\neq (0,0)} \frac{1}{z - (m\tau+n)} + \frac{1}{m\tau+n} + \frac{z}{(m\tau+n)^2} \right) \\ &= \left(\frac{1}{z+1} - \frac{1}{z}\right) \\&+ \left( \sum_{(m,n+1)\neq (0,0)}\frac{1}{(z+1) - (m\tau+n)} + \frac{(z+1)}{(m\tau+n)^2} - \sum_{(m,n)\neq 0}\frac{1}{z - (m\tau+n)} + \frac{z}{(m\tau+n)^2}  \right)
\end{align*}
Do we need some index shifting here? I am totally clueless so any help is appreciated.
 [/INST] Its derivative is $-\wp_\Gamma(z+1)+\wp_\Gamma(z)=0$, as $1\in\Gamma$ 
and $\Gamma$ is the
period lattice of $\wp_\Gamma$.
</s>