The discrete Fourier series summation is defined from 0 to N-1 because it represents a finite set of samples over one fundamental period of the signal. In contrast, the continuous Fourier series extends from negative to positive infinity, reflecting the continuous nature of the signal. For discrete signals, the summation captures the periodicity and finite duration, while the continuous case encompasses all possible frequencies. Additionally, when expressed in terms of sin(nx) and cos(nx), the summation can extend to infinity, but it remains bounded for discrete signals. Understanding these differences is crucial for analyzing signal behavior in both domains.
