Correct form of discrete-time Fourier series representation
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

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