Is it always the case that lower frequencies contribute the most in a Fourier series? Or to put it in other words, in the equation:
$$f(t)=a_0+\sum^\infty_{m=1} a_m\cos \left(\frac{2\pi mt}{T}\right) +\sum^\infty_{n=1} b_n \sin\left(\frac{2\pi nt}{T}\right) $$
Is it true that if we plot $a_m$ and $b_n$ against $m$ and $n$, the curve will be decreasing over time? If yes, why? If not, could you please provide me with some examples?
If the series converges, then the terms must approach $0$, so some smaller ones will contribute most. In some cases $|a_n|$, $|b_n|$ may increase and then decrease, or alternately increase and decrease, but the long-term trend must be to $0$ if the series is to converge.
There is, however, a periodic Dirac delta function, which isn't really a function in the sense of mapping, whose Fourier series is $$ \delta(x) = \sum_{n=-\infty}^\infty e^{inx}. $$ When $x\not\equiv2\pi n$ for any integer $n$, then this converges to $0$, but at $2\pi n$ it diverges to $\infty$.