Coefficient of Fourier Series

Question

I'm currently studying Fourier Series and this is my lecture noteFourier Series

But when im doing revision and i found this Fourier Series 2

Can i know what is the difference between these two?Why are the coefficient is different?

Answer 1

There is no difference, if you see the first case the integral is calculated over the interval $[t_o,t_o+T]$ and hence the length of the interval is $T$. On the second case the integral is calculated over the interval $[-L,L]$ and then the length is now $2L$. Dividing by $T$ or by $2L$ is analogous.