Represent 1/(x+2) as power series

Question

The title was a question so I did it by thinking r=-1-x , a=1 for geometric series but my teacher told me this didn't work. Instead, I need to factor 2 then r=-x , a=1/2 to get the correct answer.

Why doesn't mine work? I thoguht the difference came from where the point I focus on to represent the function as power series.

Mine is centered at 1 whereas another is at 0.

Answer 1

Series from 0 to Infinity of ((-1)ⁿ*xⁿ)/2ⁿ⁺¹). See attached video for an explanation of how to get this!