I am having a hard time with the following exercise :
Let $w$ be a root of unity and $u \in \mathbb{R}^{\mathbb{N}}$ a decreasing sequence. The study the nature of : $\sum w^n u_n$
It seems to me like that there a lot of cases to handle. For exemple if $u_n \to 0$, $w = -1$ it converges by the alternated serie test. Yet if $u_n \to l \ne 0$ then it doesn't converge. So first we can say that we need $u_n \to 0$ for if we want the series to converge.
Yet now I don't now where to go, since it's hard for me to see what is happening with complex numbers. Taking absolute value doesn't cut it since we only look absolute convergence in this case...