= Mercator series
{c}
{title2=$\ln(1+x)=\sum_{n=1}^\infty(-1)^{n+1}x^n/n$}
{wiki}
= Logarithmic series
{synonym}
Integrating the <geometric series> for $1/(1+x)$ from zero to $x$ gives the displayed <power series> for $|x|<1$. It has <radius of convergence> one. At $x=1$ the <alternating series test> gives <conditional convergence> to $\ln2$; at $x=-1$ the negative <harmonic series> diverges. Replacing $x$ by $-w$ gives $\ln(1-w)=-\sum_{n\ge1}w^n/n$. An expression formed by substituting a nonlinear function for $w$ need not have the same convergence range after it is re-expanded as a <Taylor series> in the new variable.
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