Writing a MacLaurin expansion and finding the radius of convergence for the function: $f(x)= \ln(x+2)$

Question

I was tasked with writing a MacLaurin expansion and finding the radius of convergence for the function:

$$f(x)= \ln(x+2)$$

I don't really know how to do this; I know that

$$\frac{1}{1+t} = -t +t^2 -t^3+\dots,\ |t|<1$$

and that when we integrate all that we get:

$$\ln|t+1| = -\frac{t^2}{2} +\frac{t^3}{3} -\frac{t^4}{4}+\dots$$

but I'm not sure as to how to link this to the question. I know that we need to integrate/differentiate the function, but where exactly is not really clear to me.

Answer 1

HINT

Note that

$$\log(x+2)=\log2+\log\left(1+\frac x 2\right)$$

and use expansion for $\log(1+y)$ at $y=0$ which converges for $|y|<1$.