What is the relationship between Fourier transformation and Fourier series? Is there any connection between Fourier transformation of a function and its Fourier series of the function? I only know the formula to find Fourier transformation and to find Fourier coefficients to find the corresponding Fourier series.

Given a locally compact abelian group $G$, one can define the character group of $G$ as the group of continuous homomorphisms $G \to S^1$. (It should actually land in $\mathbf C^\times$, but for the purpose at hand, this is good enough.)
The character group of the circle $S^1$ is isomorphic with $\mathbf Z$ (the characters are $\chi_n : \theta \mapsto e^{2\pi i n\theta}$).
On the other hand, the character group of $\mathbf R$ is isomorphic with $\mathbf R$ itself. The characters are $\chi_t : s \mapsto e^{2\pi i st}$.
It is a general principle that the characters of a locally compact group form a "basis" for the space of "nice-enough" functions on the group. Thus, periodic functions (i.e. functions on the circle) have a decomposition as sums of $e^{2\pi i n\theta}$ (Fourier series), whereas functions on $\mathbf R$ have a decomposition as Fourier integrals (inverse Fourier transform of their Fourier transform).
I'm sweeping hundreds of years of analysis under the rug (not that I know all of it), but this is the general idea.
