Logarithm of a formal power series ($\log (1+u)={\sum }_{k\ge 1}(-1{)}^{k+1}{u}^k/k$)

Over a commutative ring containing $\mathbb{Q}$, a formal power series $f=1+u$ with $u\in xR[[x]]$ has the formal logarithmEvery coefficient receives only finitely many contributions, since $u^k$ is divisible by $x^k$. No analytic convergence is involved. For example, $\log (1+x)=x-x^2/2+x^3/3-\cdots$.

The formal derivative satisfies $(\log f{)}^{\prime }=f^{\prime }/f$, and $\log (fg)=\log f+\log g$ for commuting series with constant coefficient one. This turns multiplicative identities for generating functions into additive ones. In positive characteristic, division by $k$ may fail, so this definition cannot be transferred unchanged.