Why is there an $a_0$ term in Fourier series?

Question

Fourier Series Formula

I get the reason why the Fourier Series sums up the sine and cosine function, but I do not understand what the purpose of the $a_0$ in the front? Is the $a_0$ there in case the original $f(x)$ function does not go through $(0, 0)$. I am no math genius so I am sorry if this a stupid question

Answer 1

You're quite right; you might find it an enlightening exercise to try and work out what would the Fourier series of $x \mapsto 1$ would be if you weren't allowed access to the $n=0$ term of the cosine expansion.

Answer 2

For periodic sine and cosine functions, the average value of the function over a period is zero. i.e. $$\frac 1T\int_T \sin\left(\frac{2\pi}T x\right)dx=0$$ This means if the fourier series of a function is totally expressed in terms of pure sine and cosine functions, then its average value is zero. But sometimes we have a periodic function whose average value is not zero. For example:

So here comes the mighty $a_0$ to get rid of that bias and express the rest of function in terms of sines and cosines.

Answer 3

$b_{n} sin(nx) + a_{n} cos(nx)$ put $n = 0$.