The question states, find the sum of the series
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
Am I allowed to pull the x^2 out or do I need to find a different pathway?
Taking the x2 out of the sum as you did is correct.