I am studying local fields, and to prove that the logarithm in these fields is a homomorphism he states that $\log((1+x)(1+y))=\log(1+x)+\log(1+y)$ is an identity of formal series. This question has already been aswered here, with a really clean solution by Lubin: $p$-adic logarithm is a homomorphism, formal power series proof
However, when I was trying to solve the problem, my approach was more similar to that of the OP. In fact, if I did not commit some mistake I reduced the problem to prove that for any $m,n\ge 1$, we have $\sum_{k=0}^{\infty}\frac{(-1)^{k-1}}{m+n-k}\frac{(m+n-k)!}{(m-k)!(n-k)!k!}=0$, but I don't know how to proceed from here.