Thus we arrive at the differential coefficient of f(x) as the limit of the ratio of f (x+8) - f (x) to 0 when 0 is made indefinitely small; and this gives an interpretation of nx n-1 as the derived function of xn (ï¿½ 45)ï¿½ This conception of a limit enables us to deal with algebraical expressions which assume such forms as -° o for particular values of the variable (ï¿½ 39 (iii.)).
} Nx ..., it is clear that, if x,.
=n, then we have in the development of the product a term x n, and hence that in the term Nx of the product the coefficient N is equal to the number of partitions of n with the parts I, 2, 3, ..., without repetitions; or say that the product is the generating function (G.
Observing that any factor 1/I-x l is=l+x l +x 2l +..., we see that in the term Nx the coefficient is equal to the number of partitions of n, with the parts I, 2,, ..,withh repetitions.