Definition:Exponent of Convergence

Definition
Let $\left\langle{a_n}\right\rangle$ be a sequence of nonzero complex numbers.

The exponent of convergence of $\left\langle{a_n}\right\rangle$ is the infimum of $\tau\geq0$ for which the series $\displaystyle\sum_{n\mathop=1}^\infty |a_n|^{-\tau}$ converges.

The exponent of convergence of a finite sequence is $0$.

Also see

 * Definition:Rank of Entire Function
 * Exponent of Convergence is Less Than Order, where it is shown that a function of finite order has finite exponent of convergence