Benefit to know Fourier series for image processing?
In image processing, you are always dealing with discrete data, so the transform isn't really a Fourier transform, it is a discrete Fourier transform (DFT with a sum rather than an integral).  The DFT can be understood as providing the coefficients of a Fourier series from which a periodic version of you're original image can be reconstructed.  The "checker board" image is always implied in the inverse of the DFT which would construct an infinite checker board of squares made up of your original image.

Normally the area outside the boundaries of you're original image is just ignored (treated as undefined), but if for some reason that area must be considered when reconstructing from a DFT (you transformed a sub-section of an image for example) that area will be filled with a periodic replication of of the transformed portion of you're image.

I would say it is worth the little bit of time it takes to understand Fourier series before moving on to Fourier transforms so you understand the underlying assumptions and mechanics of the transform.