Weierstrass psi function limit

1. The problem statement, all variables and given/known data

Show that

\lim_{z \to 0} z^2( \psi(z)-\psi(\frac{w_j}{2})) =1

where

\psi(z)=\frac{1}{z^2}+\sum\limits_{w \in \Omega}' \frac{1}{(z-w)^2}-\frac{1}{w^2}

where

\Omega

are the periods of

\psi(z)

2. The attempt at a solution

\lim_{z \to 0} z^2( \psi(z)-\psi(\frac{w_j}{2}))= 1 + \sum\limits_{w\in \Omega}'\frac{z^2}{(z-w)^2}-\frac{z^2}{w^2}

The last time clearly vanishes.

For the second term I can write this as

\frac{1}{(1-\frac{w}{z})^2} \to 1

z \to 0

So I get

\lim_{z \to 0} z^2( \psi(z)-\psi(\frac{w_j}{2})) =1+ \sum_{w \in \Omega}' 1

MY QUESTION

I don't really understand the summation second term here.

\sum_{n=1}^{n=n} 1 = n

right?

So there's an infinite number of

w \in \Omega

so obviously I dont want to do this, are you in affect looking at the limit '

mod \Omega

', so where the first term corresponds to

w=0

, i.e so that's us done, the limit taken for mod \Omega ?

Many thanks

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