Evaluating inverse Discrete Time Fourier Transform

Question

Say I have some discrete elements $b_{k}$ with $k=0,1,...,N-1$. I know that each element is given by a series of the form: \begin{eqnarray} b_{k} = \frac{1}{N}\sum_{j=0}^{N-1}e^{-i2\pi kj/N}.\frac{1}{1-e^{-i2\pi j/N \pm ic}} \end{eqnarray} where $c\in \mathbb{R}$, and therefore $|e^{\pm ic}|= 1$. By going to the $N\to \infty$ limit, we can write: \begin{eqnarray} b_{k}=\frac{1}{2\pi}\int_{0}^{2\pi}dx e^{-ikx}.\frac{1}{1-ae^{-ix}} && |a|\leq 1 \end{eqnarray} According to the Wikipedia page on DTFT (https://en.wikipedia.org/wiki/Discrete-time_Fourier_transform), the following holds: \begin{eqnarray} x[n]=\frac{1}{2\pi}\int_{0}^{2\pi}dx e^{+inx}.\frac{1}{1-ae^{-ix}} = a^{n}u[n] \end{eqnarray} However, in my example above, we always have $k\geq 0$, and if we use this I obtain: \begin{eqnarray} b_{k}=a^{-k}u[-k] \end{eqnarray} How does this match with the original definition of the $b_{k}$ terms? I am a bit confused with the fact that $u[-k]$ is only distinct from zero for $k\leq 0$, however this contradicts the original definition given above with $k\geq 0$. Is the DTFT formula above not correct in this case? Or how could one make this consistent?