A complex function $f(x)$ that is periodic on $[0,2L]$ can be represented as infinite sum of a complex, orthonormal exponential functions that represent the frequencies that reside in $f$.
$$ f(x) = \sum_{k=-\infty}^{\infty} c_k e^{\frac{i\pi kx}{L}} $$
Where $c_k$ are the coefficients for the frequencies and are given by
$$ c_k = \frac{1}{2L} \int_{0}^{2L} f(x) e^{\frac{-i\pi kx}{L}} dx $$
This is the Fourier series definition. However when moving to DFT we have
$$ X[k] = \sum_{n=0}^{N-1} x[n] \cdot e^{-j \cdot 2\pi \cdot k \cdot n / N} $$
$$x[n] = \frac{1}{N} \sum_{k=0}^{N-1} X[k] \cdot e^{j \cdot 2\pi \cdot k \cdot n / N}$$
In the last formula, I'm not sure how the sum from $k=0 $ to $k=N-1$ matches $ f(x) = \sum_{k=-\infty}^{\infty} c_k e^{\frac{i\pi kx}{L}}$ because it does not include negative k's
The DFT operates on a finite set of discrete samples, as a result it produces a corresponding set of discrete frequency components. Now, to understand why the DFT does not explicitly handle negative frequencies, let's focus on the exponential term in the DFT. For a given index $k$, the term $\mathrm{e}^{-i \frac{2\pi}{N} kn}$ represents a complex sinusoidal function with a frequency of $\frac{k}{N}$ cycles per sample. We can see this from Euler's formula, where the general form of a complex exponential function is given by: \begin{equation} \mathrm{e}^{i\theta} = \cos(\theta) + i \sin(\theta). \end{equation} In the context above, we have $\theta = \frac{2\pi}{N}kn$. Substituting this into Euler's formula, we get \begin{equation} \mathrm{e}^{-i \frac{2\pi}{N} kn} = \cos\left(\frac{2\pi}{N}kn\right) - i \sin\left(\frac{2\pi}{N}kn\right). \end{equation} This expression represents a complex sinusoidal function with a frequency of $\frac{k}{N}$. Consequently, the entire exponential term $\mathrm{e}^{-i \frac{2\pi}{N} kn}$ can be seen as a combination of cosine and sine functions with this frequency.
Now, if we multiply this term by the input sequence $x[n]$ and sum over all values of $n$, we essentially evaluate how much of this sinusoidal component is present in the signal at the frequency $\frac{k}{N}$. Hence, the exponential term in the DFT represents a complex sinusoidal function whose frequency is determined by the index $k$ and the length of the sequence $N$, increasing $k$ means increasing the frequency.
In the context of DFT, the frequency index $k$ ranges from 0 to $N-1$. Since the DFT assumes that the signal is periodic with a period of $N$, it only considers frequencies that are multiples of the fundamental frequency $\frac{1}{N}$. As a result, there is no explicit representation of negative frequencies in the DFT. However, the DFT implicitly handles negative frequencies due to its periodic nature. The positive frequencies are repeated periodically, and any negative frequency components can be interpreted as a wrapping or folding of the positive frequencies, as @Stefan Lafon already pointed out.
In practical terms, when you analyze the DFT output, the frequency components at indices $k$ and $N-k$ are often considered to be symmetric in magnitude but differ in phase. This symmetry reflects the periodic nature of the DFT and the implicit handling of both positive and negative frequencies within the periodic framework.