I am now considering the following infinite series.
$$ \lim_{k \to \infty} \sum_{j=1}^{k} \frac{1}{j(j+1)} \left\{\exp\left(\frac{2 \pi i}{n} \right)\right\}^{j} $$
Here, $n \geq 2$ is a natural number and $i$ is imaginary unit.
We know that this infinite series converges. How can we prove that the convergence value of this infinite series is not $0$ ? (It is not always necessary to find the limit value. Since convergence is guaranteed, it is sufficient to show that it does not converge to $0$.)
I could show in the case of $n=2$ by using partial fractional decomposition
However, in other cases, I am having trouble calculating the specifics as described above.
$\textbf{Hint:}$ By considering the series
$$\sum_{j=1}^\infty \frac{z^j}{j(j+1)}$$ and using partial fraction decomposition, we obtain that your series is equal to
$$1 - \operatorname{Log}\left(1-\exp\left(\frac{2\pi i}{n}\right)\right) + \exp\left(-\frac{2\pi i}{n}\right)\operatorname{Log}\left(1-\exp\left(\frac{2\pi i}{n}\right)\right)$$