Federico posted a question today about what equals. This is what I tried out, so let me know what you think.
He showed in class how . By differentiating both sides we get . Thus we plug 1 and get that .
Thus we have that
I tried to make sense of this combinatorially and think that it tells us that for a set such that the number of ways to choose an element of and creating a subset containing it is equal to the number of ways to choose a subset of and picking out an element from it.
For example take . For the first case say we pick the element 2, then we can create the subsets . Since we can pick 3 elements and do this for each one, we have a total of . Now for the second question we can choose the empty set but have no way of choosing an element ( 0 ways), choose a 1-set and then an element( ways), a 2-subset and then an element( ways), or a 3-subset and then an elements( ways). Thus giving us a total of 0+3+6+3=12 ways.
Cool way to use analysis and algebra to get a not so obvious(at least to me) combinatorial result.