The bandwidth (B) of the signal $x(t)$ is the range of frequencies (measured on the positive semi-axis) in which $X(\omega)$ takes values different from $0$. Very often $X(\omega)$ is different from $0$ as $-\infty$ to $\infty$. In this case the band corresponds to the frequency interval in which $X(\omega)$ is "significantly" different from $0$.
$$ x(t)= \sum_{n=-\infty}^\infty \gamma_n e^{ \frac{i2\pi nt}{T}}$$
Is possible to calculate the bandwidth B of x(t), starting from its Fourier series? And in what way?
Each term of the Fourier series, $\gamma_n e^{ i 2\pi nt/T}$, represents a harmonic of period $T/n$, hence of frequency $n/T$. If $\gamma_n=0$ for all sufficiently large $n$ (say for $|n|>N$), then you have an upper bound on frequency, namely $N/T$. In practical terms, you may decide that very small values of $\gamma_n$ are negligible and use some cutoff to determine approximate bandwidth.