# Intuition behind the Fourier series

I have a simple question:

What is the (analytical) intuition behind the Fourier series:

$f(t)=\frac{a_{0}}{2}+\sum_{k=1}^{\infty}\left(a_{k}\cos\left(kt\right)+b_{k}\sin\left(kt\right)\right)$

I have read in an article that this expression results from $f(t)=\sum_{k=1}^{\infty} d_n\cos (nt+\phi_n)$ using some trigonometric theorems. But the question is still: why can I write every function as the second expression?

• I think you should just keep going, intuition will come. This is NOT obvious at all, otherwise it would not have been such a great breakthrough. – Giuseppe Negro Nov 17 '17 at 9:35
• You can read this: betterexplained.com/articles/… – Dove Nov 17 '17 at 9:58
• Arguably, there is no good analytical intuition for being able to write $f$ as a Fourier series. When Fourier first claimed such a thing, he was banned from publishing for over a decade because the prominent Mathematicians at the time thought his claim was blatantly false. – Disintegrating By Parts Nov 17 '17 at 16:14
• Actually you can't "write every function" in general. For example, if $f(t)$ is not periodic it is impossible. Also, convergence can be tricky if $f(t)$ is not integrable. See also the Gibbs phenomenon. – Somos Nov 18 '17 at 0:19

Any function $f(x)$, between limits $a$ and $b$ can be expressed as sum of sinusoidals . This is based on the same logic as that of the Taylor/ Maclaurin series, but here for a bounded range of input.