Originally Posted by

**Ackbeet** You're confusing yourself, I think. Constructing a trigonometric Fourier series is a two-step process (depending on how you count steps):

1. Compute the coefficients $\displaystyle a_{k}$ and $\displaystyle b_{k}$, including $\displaystyle a_{0}.$ That is, compute

$\displaystyle \displaystyle{a_{0}=\frac{1}{2\pi}\int_{0}^{2\pi}f (t)\,dt,$

$\displaystyle \displaystyle{a_{k}=\frac{1}{\pi}\int_{0}^{2\pi}f( t)\,\cos(kt)\,dt,$ and

$\displaystyle \displaystyle{b_{k}=\frac{1}{\pi}\int_{0}^{2\pi}f( t)\,\sin(kt)\,dt.$