puntmat posted an interesting idea (7th August 2014) involving various functions and exploring their limit as n gets very large indeed (tends to infinity)
these resources have a similar intention
what fractions are shaded with a colour (out of the whole rectangle) at each stage?
what happens as the shape gets bigger and bigger?
all give rise to a common set (or subset) of fractions
with the same limit
the puntmat post helpfully included some animated gifs for two of these sequences:
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