Upon studying Discrete Time Fourier Transforms (DTFT), I got interested in studying the sum: $$\sum_{n=0}^{\infty} e^{-j\omega n}$$
Considering values of $\omega$ that are of the form: $2\pi \cdot r$, where $r$ is a rational number. Since $r$ is rational, i.e. $r = \frac{a}{b}$ where $a$ & $b$ are integers, we have: $$ \sum_{n=0}^{b-1} e^{-j2\pi \cdot \frac{a}{b} \cdot n} = 0$$ (Simple extension of the well known sum of roots of unity identity)
Then: $$\sum_{n=0}^{\infty} e^{-j2\pi \cdot \frac{a}{b} \cdot n} = \sum_{k=0}^{\infty} \sum_{n=kb}^{(k+1)b - 1} e^{-j2\pi \cdot \frac{a}{b} \cdot n} = \sum_{k=0}^{\infty} \ 0 = 0$$
However this is contrary to the well-known criterion of convergence of a geometric series of infinite terms $\sum_{n=0}^{\infty} a^n$, which requires $|a| < 1$.
Am I doing something wrong?
You can regroup terms like you do only if you have absolute convergence.
Here is a simpler example to understand what goes wrong :
Clearly, $\displaystyle\sum_{n\geq 0}(-1)^n$ diverges (the partial sums are alternatively $1$ and $0$).
Now $$\sum_{m\geq 0}((-1)^m+(-1)^{m+1})=\sum_{m\geq 0}0=0,$$ so if you regroup terms, you have a convergent series.