Suppose I have the following definition: $$\sum_{k=0}^N m_k\zeta^{1-k},$$
where $m_k$ is an integer and $\zeta$ is a complex number $\zeta=e^{i\theta}=\cos \theta + i\sin\theta$ with $\theta = 0\cdots2\pi$. This can also be expanded as:
$$\sum_{k=0}^N m_k\zeta^{1-k}=\sum_{k=0}^N m_k\zeta\zeta^{-k}.$$
But, why is it not correct to write
$$\sum_{k=0}^N m_k\zeta^{1-k}=\sum_{k=0}^N m_k\zeta + \sum_{k=0}^N m_k\zeta^{-k}?$$
My calculation suggests that the real part of latter always gives me twice as the first two.
Can anyone explain?