linguistic:voice_change
Dirichlet series

An extension of the area of convergence can be obtained by rearranging the original series. The series

converges for Re(s) > 0, while

converge even for Re(s) > −1. In this way, the area of convergence can be extended to Re(s) > − k for any negative integer − k.

The recurrence connection is clearly visible from the expression valid for Re(s) > −2 enabling further expansion by integration by parts.