Convergent series/Not absolutely convergent/Alternating series of unit fractions/Value/Example

A convergent series does not in general converge absolutely, the converse of fact does not hold. Due to the Leibniz criterion, the alternating harmonic series

$\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\dots$

converges, and its sum is $\ln 2$, a result we can not prove here. However, the corresponding absolute series is just the harmonic series, which diverges due to example.