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

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

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.