a/b + a(a+1)/b(b+1) + a(a+1)(a+2)/b(b+1)(b+2)+...........................=?????????
first term a/b
second term: first term times (a+1)/(b+1)
and so it continues.
Looks a lot like a factorial.working from the bottom up.
Off hand I don't know but I am sure it is amenable to reason.
I see a choice of attacking it with sigma notation or tearing it apart with more mundane algebra, or possibly both.
Then again, maybe this a run-of-the-mill calculus topic, which at this time is not foremost in my mind. But I found this at Elementary & Middle School site. It does not really make a lot of difference, except that I would not expect to see this in the middle schools around here. This is just the sort of thing I want to impart to my daughter at this time: straightforward algebraic logic.
I think I would try be seeing it as one series divided by another series, since both the numerator and the denominator are increasing in the same manner.
Really what you need to do is determine a function f(n) that describes the growth of either the numerator or denominator and then use "a" and "b" where you formerly had "n".
So you will have sum of f(a)/f(b) from i = 1 to whatever.
I don't want to work on this right now and it is probably far too late for you. But I would be keenly interested in what you came up with if anything. Surely there is someone who would be able to solve this so-called pure math problem as if by rote. I would have to work at it for a while.
Thanks for your inquiry, it is something to keep my mind on math.