= Logarithm of a formal power series
{title2=$\log(1+u)=\sum_{k\ge1}(-1)^{k+1}u^k/k$}
= Formal power series logarithm
{synonym}
Over a <commutative ring> containing $\mathbb Q$, a <formal power series> $f=1+u$ with $u\in xR[[x]]$ has the formal logarithm
$$
\log f=\sum_{k\ge1}\frac{(-1)^{k+1}}k u^k.
$$
Every 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 $(\log f)'=f'/f$, 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.
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