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The question states, find the sum of the series

Sigma(0,inf) (n+2)x^n

I multitplied the series by x/x so that now I have

1/x Sigma(0,inf) (n+2)x^(n+1)

I see that (n+2)x^(n+1) is the derivative of x^(n+2) so I write

1/x Sigma(0,inf) d/dx x^(n+2)

Now I am stuck though, because I don't know how to represent x^(n+2) in another way.

I wanted to take an x^2 out and leave only x^n inside the sum so that I could write it as a geometric series but I'm not sure if the derivative prevents that or not, ie

x^2/x Sigma(0,inf) d/dx x^n

which would then be easy to solve and would equal

x/(1-x)^2

Am I allowed to pull the x^2 out or do I need to find a different pathway?

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Taking the x^{2} out of the sum as you did is correct.