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This question already has an answer here:

Any continued fraction that does not terminate or repeat can't be rational or a quadratic irrational. It is not hard to write something that does not fit these two categories.

Can we still get a closed form for something like: $$ [1,2,3,4,5, \dots ] = 1 + \cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{\dots}}}$$

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marked as duplicate by GEdgar, Did, Rory Daulton, PVAL-inactive, J. W. Perry Apr 15 '15 at 0:18

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