Correct form of discrete-time Fourier series representation

Question

As I see in this slides, Fourier series representation for discrete-time signal $s[n]$ with period $N$ is $\sum_{k = 0}^{N-1} c_k e^{j2\pi k n / N}$

According to Wiki, Fourier series representation for continuous-time signal $s(x)$ with period $P$ is $s_{\infty}(x) = \sum_{n = -\infty}^\infty c_n e^{j 2\pi n x / P}$

My question is that why in discrete-time Fourier series representation, they only sum up from $0$ to $N - 1$ instead of from $-\infty$ to $\infty$ ? Is the slides above gives a wrong formula ?

P.s: in my opinion, the more terms we have, the more accurate our Fourier series is

Answer 1

Because the discrete-time basis functions $e^{j2\pi kn/N}$ are not only $N$-periodic in $n$ (as is the signal $s[n]$), but also in $k$. So adding terms for $k\ge N$ does not add anything new, simply because

$$e^{j2\pi (k+N)n/N}=e^{j2\pi kn/N}e^{j2\pi n}=e^{j2\pi kn/N}$$