Sequences and Series/Dirichlet‒Hurwitz series
Definition (Dirichlet‒Hurwitz series):
Let be a function, and let . The Dirichlet‒Hurwitz series associated to and is the function of given by the series
- .
Definition (abscissa of absolute convergence of Dirichlet‒Hurwitz series):
Let be a function, and let . Suppose that there exists a number such that
converges whenever and diverges whenever . Then is called the abscissa of absolute convergence of the Dirichlet‒Hurwitz series associated to and .
Proposition (existence of abscissa of absolute convergence of Dirichlet‒Hurwitz series):
Let be a function, and let . Suppose that