Quiz question: Can you solve this natural number sum problem?

[roger]

Hi,

I'm not sure how to tackle this quiz question I came across :

Let P(n) be the number of ways of writing a natural number n as the sum of smaller natural numbers eg. P(4)=5 as 4=4=3+1=2+2=2+1+1=1+1+1+1

I must show that the sigma P(n) = 1/(1-x)*1/(1-x^2)*1/(1-x^3)...


www.maths.bris.ac.uk/~maxmg/docs/problems.pdf[/URL]

In fact, I may not have even understood the problem, so here is the source where this came from.

thanks

Roger

 

[matt grime]

i'm pleased someone at least will try some of those questions. the question is no harder than knowing why the binomial expansion (x+y)^n is correct.

 

[uart]

I'm unclear about two things in that question.

1. What range is P(n) summed over.

2. What is x.

 

[roger]

well it doesn't state it so I guess infinity.

 

[matt grime]

The question is clearly wrong (I will correct the source). It is the coefficients you must find. ie

$$\sum_{n=0}^{\infty}P(n)x^n$$

not the sum over n of P(n) which makes no sense.

it is an exercise in formal power series.

 

[roger]

matt, is that question linked to question 7 or is it a separate question ?

 

[roger]

I also gave you a private message, I wanted to know the equation which governs p(n) ?

 

[matt grime]

reread the question on the PDF. it obviously makes no sense since summing P(n) over n is adding up an infinite number of strictly positive integers, and the RHS is a formal power series in x. I have now given you the corrected left hand side. 'The formula' for P(n) (if there is a meaningful one, in n, that isn't simply a tautology) is of no importance.

 

[roger]

I understand that its not relevant to this particular question , but it just got me thinking, if there was a way to prove the formula which I think was due to ramanujan of india.

And in what way is it a tautology ?

 

[roger]

and why didnt you specify on the original what k and n summed up to or the product in the case of k ?

does this imply its infinite ?

 

[matt grime]

sum is over all n, that is implicit and shouldn't need to be stated (the things on the other side of the equation are obviously infinitely long power series). i may be breaking the rules in your opinion, but i am allowed to since i know what the rules are and i'd expect anyone doing the question to be able to work out what it meant. in any case i can't recall exactly what i did use as notation, but mistakes do happen, as we have seen. it might be good for you to bear in mind that writing maths (and other things) is hard and will often produce mistakes that you don't notice. sorry, will try harder to be infallible next time. (and no this isn't taking criticism badly this is me going 'well of course it means that, what else could it mean? grr, where has common sense gotten to these days?')