It seems as though the minus sign is left out every so often when writing down Fourier series:
$$f(x) = \sum_{-\infty}^\infty c_k \mathrm e^{-ikx}$$
versus
$$f(x) = \sum_{-\infty}^\infty c_k \mathrm e^{ikx}$$
In the first instance,
$$\mathrm e^{ikx}=\cos kx +i\sin kx$$
while
$$\mathrm e^{-ikx}=\cos kx -i\sin kx$$
It makes intuitive sense that the minus sign in the last equation can easily be absorbed into the coefficients.
But what is the reason to use the minus sign? And why are they equivalent?
The Fourier transform and the inverse Fourier transform always come in pairs. So you transform from $x$ to $k$ and back. Which one has the negative sign is just a convention.