I already know that it is real analytic in 1, with radius of convergence 1 and power series expansion:

$\displaystyle ln(x)=\Sigma_{n=1}^\infty \frac{(-1)^{n+1}}{n} (x-1)^n$

for $\displaystyle x \in (0,2)$

Now what?

I think that I can show that ln is real analytic for all points of (0,2). But I'd like to prove it for $\displaystyle (0, + \infty)$.

Some thoughts?