Coefficients of the Fourier Series expansion of physical signals having finite power
A finite power periodic signal will have the following property:

$$ P_x = \frac{1}{T_0} \int_{0}^{T_0} |x(t)|^2 dt  &lt; \infty \tag{1} $$

where $P_x$ is the power averaged over a period of the signal. Then from Parseval's theorem, the total power can also be shown to be (for a real signal I'm assuming):

$$P_x = |a_0|^2+2 \sum_{n=1}^{\infty} |a_n|^2 &lt; \infty \tag{2}$$ 

where the $a_n$'s are the continuous-time Fourier series coefficients for $x(t)$.

From the theory of power series (calculus), it's known that in order for (2) to converge (sum being less than infinity) a necessary condition is

$$ \lim_{n \to \infty} |a_n|^2 = 0 \tag{3}$$

(3) implies therefore that, for finite power periodic signal's CTFS coefficients $a_n$ also goes to zero as $n$ goes to infinity.