You could expand ln(1+x) in its Taylor series around x=0.
For this, you need all the derivative of ln(1+x) evaluated at x=0.
Let f(x)=ln(1+x). (note first that f(0) = 0).
f'(x) = 1/(1+x).
f''(x) = -1/(1+x)^2.
f'''(x) = 2/(1+x)^3.
f''''(x) = -6/(1+x)^4
and so on.
In general, we have that the n'th derivative of f is (which you may prove by induction). Then
and so the Taylor series is:
which converges for (except for ).