I know the taylor series of $\log(1+x)$. However I don’t understand how to find the convergence for $x>1$ and divergence if $x<1$.
We have (i) convergence if $|x|\lt 1$, and divergence if $|x|\gt 1$. This can be done by using the Ratio Test.
We also have (ii) convergence at $x=1$ and divergence at $x=-1$. For $x=1$, we have an alternating series. For $x=-1$, we get a close relative of a familiar divergent series.