I am studying for an exam in Differential Equations, and one of the topics I should know about is Fourier series. Now, I am using Boyce 9e, and in there I found the general equation for a Fourier series:
$\frac{a_0}{2} + \sum_{m=1}^{\infty} (a_m cos\frac{m \pi x}{L} + b_m sin\frac{m \pi x}{L})$
I also found the equations to calculate the coefficients in terms of n, where n is any real integer:
$a_n = \frac{1}{L} \int_{-L}^{L} f(x) cos\frac{n \pi x}{L}dx$ $b_n = \frac{1}{L} \int_{-L}^{L} f(x) sin\frac{n \pi x}{L}dx$
I noticed that the coefficients are calculated in terms of n, but are used in the general equation in terms of m. I also noticed that at the end of some exercises in my book, they convert from n to m. So my question is: what is the difference between n and m, and why can't I calculate my coefficients in terms of m directly? Why do I have to calculate them in terms of n, and then convert them? I hope that some of you can help me out!