Definition:Exponent of Convergence

Definition
Let $\sequence {a_n}$ be a sequence of nonzero complex numbers.

The exponent of convergence of $\sequence {a_n}$ is the infimum of $\tau \ge 0$ for which the series $\displaystyle \sum_{n \mathop = 1}^\infty \size {a_n}^{-\tau}$ 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