Calculating the complex coefficient for the dirac comb

Question

Im trying to calculate the complex coefficient $c_n$ for the dirac comb but im stuck in this step which leads to 2/T instead of 1/T: so the sum vanished because $t \in [0, T]$ so k must be either $ 0; 1$, here are the calculations: $c_n = \frac{1}{T} \int_{0}^T \omega_T (t) exp(-i \frac{2 \pi n}{T} t) dt = \frac{1}{T} \int_{0}^T \sum_{k=-\infty}^\infty \delta(t - kT) exp(-i \frac{2 \pi n}{T} t) dt = \frac{1}{T} \sum_{k=-\infty}^\infty \int_{0}^T \delta(t - kT) exp(-i \frac{2 \pi n}{T} t) dt = \frac{1}{T} (1+ exp(- i \frac{2 \pi n}{T} * T)) = 2/T $