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I have to solve the recurrence relation: $$a_n=a_{n-1}+n\quad(n\geq1), \quad a_0=0$$ with generating function. The final result I think should be: $a_n=\frac {n(n+1)}2$, but I don't know how to get it from the function. I tried with partial fraction decomposition, but the problem is that I don't know how to deal with denominator. This is what I did: $$\sum_{n=1}^{+\infty}a_nx^n=\sum_{n=1}^{+\infty}a_{n-1}x^n+\sum_{n=1}^{+\infty}nx^n$$$$f(x)-a_0=xf(x)+\frac x{(1-x)^2}$$$$f(x)-xf(x)=\frac x{(1-x)^2}$$$$f(x)(1-x)=\frac x{(1-x)^2}$$$$f(x)=\frac x{(1-x)^3}$$So I have the generating function but I don't know how to get to the result from here.

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    $\begingroup$ Better if you write out what you did & see if we can get you over your difficulty. $\endgroup$ – Gerry Myerson Jun 11 at 11:04
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    $\begingroup$ Hint: Try applying the Binomial Theorem to $(1-x)^{-3}$ $\endgroup$ – awkward Jun 11 at 12:11
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    $\begingroup$ Edited with the steps I did $\endgroup$ – Diego Di Marzo Jun 11 at 12:34
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Hint:

By the generalized binomial theorem,

$$(1-x)^{-3}=1+3x+\frac{3\cdot4}{2}x^2+\frac{3\cdot4\cdot5}{3!}+\frac{3\cdot4\cdot5\cdot6}{4!}+\cdots\frac{(n+2)!}{2\,n!}x^n\cdots$$


Also by Taylor $$\frac x{(1-x)^3}=\frac{1-(1-x)}{(1-x)^3}=\frac1{(1-x)^3}-\frac1{(1-x)^2} \\=1+3x+3\cdot4\frac{x^2}2+3\cdot4\cdot5\frac{x^3}{3!}+\cdots-1-2x-2\cdot3\frac{x^2}2-2\cdot3\cdot4\frac{x^3}{3!}+\cdots$$

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    $\begingroup$ I never heard about Binomial Theorem, is this the only way? $\endgroup$ – Diego Di Marzo Jun 11 at 12:37
  • $\begingroup$ @DiegoDiMarzo: it is the Generalized Binomial Theorem. $\endgroup$ – Yves Daoust Jun 11 at 12:41
  • $\begingroup$ Got it, can I ask you if this is the only way to have the result? Because I never used it with this kind of problem. $\endgroup$ – Diego Di Marzo Jun 11 at 12:47
  • $\begingroup$ @DiegoDiMarzo: most probably not. $\endgroup$ – Yves Daoust Jun 11 at 12:49
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    $\begingroup$ There's always another way to do things, but this is the standard way that everyone teaches and is in all the textbooks. $\endgroup$ – Gerry Myerson Jun 11 at 12:57

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