# 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