Fourier Series Representation of Continuous-Time Periodic Signals
When you are given a continuous-time periodic signal $x(t)$ and you want to find out the corresponding CTFS (continuous-time Fourier series) coefficients $a_k$ associated with $x(t)$, then you use the analysis equation; i.e, analyse $x(t)$ to find out $a_k$ use

$$ \boxed{ a_k  = \frac{1}{T} \int_{0}^{T} x(t) e^{-j \frac{2 \pi}{T} k t} dt  }\tag{1}$$

On the other hand, when you are given a set of CTFS coefficients $a_k$ and you want to obtain the corresponding continuous-time periodic signal $x(t)$, then you use the synthesis equation; i.e., sum up exponentials weighted by $a_k$ and create $x(t)$ as in

$$ \boxed{ x(t) = \sum_{k=-\infty}^{\infty} a_k e^{j \frac{2\pi}{T} k t } } \tag{2}$$