Fourier Series from m=1 to infinity

[Dr_Pill]

Simple question;

Why isn't it $\sum$ a_m (from m=1 to infinity)

Thanks in advance.

 

[jbunniii]

Simple question;

Why isn't it $\sum$ a_m (from m=1 to infinity)

Thanks in advance.

In the first line, $f(x)$ was replaced with its Fourier series expansion:
$$f(x) = a_0 + \sum_{n=1}^{\infty}(a_n \cos nx + b_n \sin nx)$$
The $n$ is the index over which the sum is taken. The name of this index is arbitrary. In principle you could use $m$ instead:
$$f(x) = a_0 + \sum_{m=1}^{\infty}(a_m \cos mx + b_m \sin mx)$$
However, in this case that would cause a conflict because $m$ is already in use in the integrand on the left hand side:
$$\int_{-\pi}^{\pi} f(x) \cos mx dx$$