Quiz question

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

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Homework Helper
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.

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I'm unclear about two things in that question.

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

2. What is x.

roger
well it doesnt state it so I guess infinity.

Homework Helper
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.

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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) ?

Homework Helper
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 ?