Thread: question on progression

sum of the given series upto n terms:

a/b + a(a+1)/b(b+1) + a(a+1)(a+2)/b(b+1)(b+2)+...........................=?????????

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.

Bye.

Originally Posted by Bradley

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.

Bye.

thak you for your suggestion , i thav already used sigma notation followed by partial fraction and tried to convert it into a kind of series but was unsuccessful.

Over summer I will probably get a bit quicker at this sort of thing because I have a text on analysis to go through with my kid, as well as some texts to pick up a nice segue into baby calculus.

I know that:

sigma f(a)/f(b) from i = 1 to n

is exceedingly simple, and I hope you do, too. What I would propose below is not elegant but often works quite well for those among us who do not see the simplicity.

In the past I have had good results from this approach even before I had a clue as to what calculus was all about: Iteration, sometimes called the brute force method (not because it is particularly powerful but because it seem rather brutish compared to an elegant logical solution).

Simply run the various parts of the series in a spreadsheet, then (1) discern the most elegant function to generate each term, (2) analyze the trends of the mega function (i.e., g(x)=f(a)/f(b)), and (3) attempt to find a general formula for any particular term, any sum of terms, and if the series converges, the sum at infinity, from the data.

For those of us who have enough cash to get some math software it is possible to get the software to answer the question. But that is probably nowhere near as useful as analyzing the thing with whatever knowledge of algebra one has, at least as far as math education is concerned. It certainly would not be time wasted if one's goal was to understand the math.