Geometric-series bound for the logarithmic remainder
= Geometric-series bound for the logarithmic remainder
{title2=$-\log(1-u)-u<u^2/[2(1-u)]$}
For $0<u<1$, the <power series> for the <logarithm> gives $-\log(1-u)-u=\sum_{n=2}^\infty u^n/n<\frac12\sum_{n=2}^\infty u^n=u^2/[2(1-u)]$. The inequality is strict because $1/n<1/2$ for every $n\geq3$.