Matusubara sum contradiction?

Question

In many textbooks, the following fermionic Matsubara sum is given as a useful identity: $$T\sum_{\omega_n}\frac{1}{i\omega_n-\epsilon}=\frac{1}{e^{\epsilon/T}+1},$$ where $\omega_n=n\pi T$ for all odd $n$. However, when $\epsilon=0$, the LHS vanishes while the RHS equals 1/2. Did I misunderstand the identity?