It is an exercise about Eisenstein Series $G_2(\tau)$, to prove that $$(G_2[\gamma]_2)(\tau)=G_2(\tau)-{2\pi i\over\tau}$$ where $\gamma=\begin{pmatrix} 0&1\\ -1 &0 \end{pmatrix}$
I just cannot show that $$\sum_{d\in\mathbb{Z}}\sum_{c\ne0}\frac{1}{(c\tau+d)(c\tau+d+1)}=-{2\pi i\over \tau}$$ I tried $$\lim_{N\to\infty}\sum_{d=-N}^{N-1}\sum_{c\ne0}\frac{1}{(c\tau+d)(c\tau+d+1)}=\lim_{N\to\infty}\sum_{c\ne0}({1\over c\tau-N}-{1\over c\tau+N})$$ and to use the equality $$\pi\cot\pi N/\tau={\tau\over N}-\tau\sum_{c=1}^{\infty}({1\over c\tau-N}-{1\over c\tau+N})$$ but $\displaystyle\lim_{N\to\infty}\pi\cot\pi N/\tau$ doesn't exist.