A harmonic progression is a progression formed by taking the reciprocals of an arithmetic progression (so an example is 1/1, 1/2, 1/3, 1/4 …). When tutoring mathematics at Oxford, Charles Dodgson had a favorite example to illustrate this:

According to him, it is (or was) the rule at Christ Church that, if an undergraduate is absent for a night during term-time without leave, he is for the first offence sent down for a term; if he commits the offence a second time, he is sent down for two terms; if a third time, Christ Church knows him no more. This last calamity Dodgson designated as ‘infinite.’ Here, then, the three degrees of punishment may be reckoned as 1, 2,

infinity. These three figures represent three terms in an ascending series of Harmonic Progression, being the counterparts of 1, 1/2, 0, which are three terms in a descending Arithmetical Progression.

— Lionel A. Tollemache, “Reminiscences of ‘Lewis Carroll,'” *Literature*, Feb. 5, 1898