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I'm given: $a_n= n$ for $n \geqslant 0$.

I'm quite good at recursive generating functions, but I haven't came across a simpler one like this, so I'm sure I'm just overlooking something really basic.

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What function are you talking about? –  Ian Coley Apr 18 '13 at 16:48
    
That's literally what I'm given: The sequence a_n defined by a_n = n for n >= 0. Explain how this function is derived. (And I'm asked to give the generating function of that sequence.) –  Doug Smith Apr 18 '13 at 16:50
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According to Wikipedia, the ordinary generating function of a sequence an is $$ G(a_n,x)=\sum_{n=0}^\infty a_nx^n. $$ In your case, $G(a_n,x)=\sum_{n=0}^\infty nx^n$. –  Ian Coley Apr 18 '13 at 16:53
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The generating function should be just $\sum \limits_{n=0} n\cdot x^n$... No? –  Harold Apr 18 '13 at 16:54
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2 Answers

The generating function is, by definition $$ g(x) = \sum_{n=0}^\infty a_n x^n = \sum_{n=0}^\infty n x^n $$ The sum can be evaluated as follows: $$ g(x) = \sum_{n=0}^\infty n x^n = 0 + \sum_{n=1}^\infty n x^n \stackrel{n=m+1}{=} \sum_{m=0}^\infty (m+1) x^{m+1} = x \sum_{m=0}^\infty (m+1) x^{m} = x \left( \sum_{m=0}^\infty m x^{m} +\sum_{m=0}^\infty x^{m} \right) = x \left( g(x) + \frac{1}{1-x} \right) $$ Now solve for $g(x)$ to get $$ g(x) = \frac{x}{(1-x)^2} $$

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+1 for simplicity –  Sasha Apr 18 '13 at 16:57
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Wouldn't it be $A(x)=\sum_{i=1}^\infty ix^i$? To get this in a closed form, we have $A(x)=x\frac d{dx}\sum_{i=1}^\infty x^i=x\frac d{dx}\frac x{1-x}$

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you beat me to it :) –  gt6989b Apr 18 '13 at 16:55
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