A doubly infinite sequence is a sequence 
indexed by all integers,
so its indices extend without bound in both directions. Equivalently, it is a function 
from the integers to a
set 
,
with 
. It has neither a first nor a last index (Duke
University).
The sum of the terms of a doubly infinite sequence is a two-sided series, which converges when its two tails converge separately. The discrete-time Fourier transform and the almost Mathieu operator act on doubly infinite sequences.
