The general form of a DFS is:
Discrete Fourier series
![{\displaystyle {\begin{aligned}s[n]=\sum _{k}S[k]\cdot e^{i2\pi {\frac {k}{N}}n},\quad n\in \mathbb {Z} ,\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/964172fcbe5e913132fd3a486124dd73e66ea834) |
|
(Eq.1) |
which are harmonics of a fundamental frequency 
for some positive integer 
The practical range of 
is ![{\displaystyle [0,\ N-1],}](//wikimedia.org/api/rest_v1/media/math/render/svg/52c76144172d5816b015e673c34ce7e65333e878)
because periodicity causes larger values to be redundant. When the ![{\displaystyle S[k]}](//wikimedia.org/api/rest_v1/media/math/render/svg/5b988b0ce17df34ae3c6aad5b6f9b5ab0f5fb2f5)
coefficients are derived from an 
-length DFT, and a factor of 
is inserted, this becomes an inverse DFT.[1]: p.542 (eq 8.4) [2]: p.77 (eq 4.24) And in that case, just the coefficients themselves are sometimes referred to as a discrete Fourier series.[3]: p.85 (eq 15a)
Example
A common practice is to create a sequence of length from a longer ![{\displaystyle s[n]}](//wikimedia.org/api/rest_v1/media/math/render/svg/ffe878c033cc70d1aa3ffbd156394791aa8dffde)
sequence by partitioning it into -length segments and adding them together, pointwise.(see DTFT § L=N×I) That produces one cycle of the periodic summation:
![{\displaystyle s_{_{N}}[n]\ \triangleq \ \sum _{m=-\infty }^{\infty }s[n-mN],\quad n\in \mathbb {Z} .}](//wikimedia.org/api/rest_v1/media/math/render/svg/b128330181d4ed8a2e3147b4945078742203cfde)
Because of periodicity, 
can be represented as a DFS with unique coefficients that can be extracted by an -length DFT.[1]: p 543 (eq 8.9) : pp 557-558 [2]: p 72 (eq 4.11)
![{\displaystyle {\begin{aligned}&s_{_{N}}[n]={\frac {1}{N}}\sum _{k=0}^{N-1}S_{N}[k]\cdot e^{i2\pi {\tfrac {k}{N}}n};\quad n\in \mathbb {Z} &&{\text{DFS}}\\&S_{N}[k]\triangleq \sum _{n=0}^{N-1}s_{_{N}}[n]\cdot e^{-i2\pi {\tfrac {k}{N}}n};\quad k\in [0,N-1]&&{\text{DFT}}\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/c1c69f1e124180638ce0c93720845202340fa790)
The coefficients are useful because they are also samples of the discrete-time Fourier transform (DTFT) of the sequence:
Here, 
represents a sample of a continuous function 
with a sampling interval of 
and 
is the Fourier transform of 
The equality is a result of the Poisson summation formula. With definitions ![{\displaystyle s[n]\triangleq T\ s(nT)}](//wikimedia.org/api/rest_v1/media/math/render/svg/d1d5b885ecefa3c8d5398e1d9954be0fd73fe04e)
and ![{\displaystyle S[k]\triangleq S_{1/T}\left({\tfrac {k}{NT}}\right)}](//wikimedia.org/api/rest_v1/media/math/render/svg/08c0f539f442c80c58cdabe2c613aedb9cb9d091)
:
![{\displaystyle S[k]=\sum _{n=-\infty }^{\infty }e^{-i2\pi {\tfrac {k}{N}}n}\ s[n];\quad k\in \mathbb {Z} .}](//wikimedia.org/api/rest_v1/media/math/render/svg/c6acf24246be330e3803fb3688d98fb324132b3c)
Due to the -periodicity of the 
kernel, the summation can be "folded" as follows:
![{\displaystyle {\begin{aligned}S[k]&=\sum _{m=-\infty }^{\infty }\left(\sum _{n=0}^{N-1}e^{-i2\pi {\tfrac {k}{N}}n}\ s[n-mN]\right)\\&=\sum _{n=0}^{N-1}e^{-i2\pi {\tfrac {k}{N}}n}\underbrace {\left(\sum _{m=-\infty }^{\infty }s[n-mN]\right)} _{s_{_{N}}[n]}\\&=S_{N}[k].\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/997131b4c4cd0cc8a19dae597849ef8731b1b366)
![{\displaystyle {\begin{aligned}s[n]=\sum _{k}S[k]\cdot e^{i2\pi {\frac {k}{N}}n},\quad n\in \mathbb {Z} ,\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/964172fcbe5e913132fd3a486124dd73e66ea834)
![{\displaystyle [0,\ N-1],}](http://wikimedia.org/api/rest_v1/media/math/render/svg/52c76144172d5816b015e673c34ce7e65333e878)
![{\displaystyle S[k]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/5b988b0ce17df34ae3c6aad5b6f9b5ab0f5fb2f5)
![{\displaystyle s[n]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/ffe878c033cc70d1aa3ffbd156394791aa8dffde)
![{\displaystyle s_{_{N}}[n]\ \triangleq \ \sum _{m=-\infty }^{\infty }s[n-mN],\quad n\in \mathbb {Z} .}](http://wikimedia.org/api/rest_v1/media/math/render/svg/b128330181d4ed8a2e3147b4945078742203cfde)
![{\displaystyle {\begin{aligned}&s_{_{N}}[n]={\frac {1}{N}}\sum _{k=0}^{N-1}S_{N}[k]\cdot e^{i2\pi {\tfrac {k}{N}}n};\quad n\in \mathbb {Z} &&{\text{DFS}}\\&S_{N}[k]\triangleq \sum _{n=0}^{N-1}s_{_{N}}[n]\cdot e^{-i2\pi {\tfrac {k}{N}}n};\quad k\in [0,N-1]&&{\text{DFT}}\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/c1c69f1e124180638ce0c93720845202340fa790)
![{\displaystyle s[n]\triangleq T\ s(nT)}](http://wikimedia.org/api/rest_v1/media/math/render/svg/d1d5b885ecefa3c8d5398e1d9954be0fd73fe04e)
![{\displaystyle S[k]\triangleq S_{1/T}\left({\tfrac {k}{NT}}\right)}](http://wikimedia.org/api/rest_v1/media/math/render/svg/08c0f539f442c80c58cdabe2c613aedb9cb9d091)
![{\displaystyle S[k]=\sum _{n=-\infty }^{\infty }e^{-i2\pi {\tfrac {k}{N}}n}\ s[n];\quad k\in \mathbb {Z} .}](http://wikimedia.org/api/rest_v1/media/math/render/svg/c6acf24246be330e3803fb3688d98fb324132b3c)
![{\displaystyle {\begin{aligned}S[k]&=\sum _{m=-\infty }^{\infty }\left(\sum _{n=0}^{N-1}e^{-i2\pi {\tfrac {k}{N}}n}\ s[n-mN]\right)\\&=\sum _{n=0}^{N-1}e^{-i2\pi {\tfrac {k}{N}}n}\underbrace {\left(\sum _{m=-\infty }^{\infty }s[n-mN]\right)} _{s_{_{N}}[n]}\\&=S_{N}[k].\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/997131b4c4cd0cc8a19dae597849ef8731b1b366)