Power Series to Functional Notation?

Question

I need some help with my calculus homework. I'm trying to figure out the series representation for f(x) = ln(x+4) about c = -2. I know that you have to first take the derivative (1/x+4) so as to make it a rational function but then I don't know how to manipulate it so as to make it geometric form. I also have to specify the domain.

Answer 1

1/(x+4) = 1/[(x+2) + 2] = ½ / [½(x+2) + 1]

Now you can write this as a geometric series:

½ / [½(x+2) + 1] = ½ {1 + ½(x+2) + [½(x+2)]² + [½(x+2)]³ + ...}

provided | ½(x+2) | < 1 (otherwise the nth term doesn't go to 0). That tells you the domain.