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I'm reading page 10 of Schaum's ouTlines "Signals and Systems" by Hwei P. HSU, and I don't understand why this factor in the complex exponential sequence equals 1.

Consider the complex exponential sequence with frequency ($\Omega_{0} + 2\pi k$), where $k$ is an integer:

$e^{j(\Omega_{0} + 2\pi k)n} = e^{j\Omega_{0}n}e^{j2\pi kn} = e^{j\Omega_{0}n}$ $\quad$ $(1.56)$

since $e^{j2 \pi k n} = 1$.

My question: why does $e^{j2 \pi k n} = 1$ which, to my understand, would imply that $k$ is not just any integer but $0$ for this to be true since $j$ can't be $0$ and $n$ can't be $0$ because then $e^{j\Omega_{0}n}$ would also equal $1$.

The texts proceeds:

From Eq. $(1.56)$ we see that the complex exponential sequence at $\Omega_{0}$ is the same as that at frequencies ($\Omega_{0} \pm 2\pi$), ($\Omega_{0} \pm 4\pi$), and so on. Therefore, in dealing with discrete-time exponentials, we need only consider an interval of length $2\pi$ in which to choose $\Omega_{0}$.

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$\exp(j2\pi kn) = \cos(2\pi kn) + j\sin(2\pi kn)$, where $\cos(2\pi kn) = 1$ for $kn \in \mathbb{Z}$, i.e., integer. And $\sin(2\pi kn) = 0$ for $kn \in \mathbb{Z}$. Thus $\exp(j2\pi kn) = \cos(2\pi kn) + j\sin(2\pi kn) = 1$ for $kn \in \mathbb{Z}$

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