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).

Then

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 ).