I'm trying to find the generating function and the closed form for the generating form for this sequence:
$0,1,-2,4,-8,16,-32,64...$
I've tried the following:
I think it's an index shift so that's why the generating function is: $a_n= $?
What about the closed form? Can you please tell how I solve this, and not only the result.
$\endgroup$ 71 Answer
$\begingroup$The generating function is $$ g(x)=0+1\cdot x-2x^2+4x^3-8x^4+16x^5-\ldots. $$ Observe that each coefficient starting with the coefficient of $x^2$ is $-2$ times the coefficient of the previous term. This suggests the idea of multiplying $g(x)$ by $-2x$ and subtracting the result from $g(x).$ If you do this, you will find that all terms cancel but one. So you have $g(x)-(-2xg(x))$ equals the leftover term. You should then be able to solve for $g(x)$ algebraically.
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