Doubly Infinite Sequences
Normally, the term infinite sequence refers to a sequence which is infinite in one direction, and finite in the other—the sequence has a first element, but no final element (a singly infinite sequence). A doubly infinite sequence is infinite in both directions—it has neither a first nor a final element. Singly infinite sequences are functions from the natural numbers (N) to some set, whereas doubly infinite sequences are functions from the integers (Z) to some set.
One can interpret singly infinite sequences as elements of the semigroup ring of the natural numbers, and doubly infinite sequences as elements of the group ring of the integers . This perspective is used in the Cauchy product of sequences.
Read more about this topic: Sequence
Famous quotes containing the words doubly and/or infinite:
“For the most part, we are not where we are, but in a false position. Through an infirmity of our natures, we suppose a case, and put ourselves into it, and hence are in two cases at the same time, and it is doubly difficult to get out.”
—Henry David Thoreau (1817–1862)
“Each man has his own vocation. The talent is the call. There is one direction in which all space is open to him. He has faculties silently inviting him thither to endless exertion. He is like a ship in the river; he runs against obstructions on every side but one; on that side all obstruction is taken away, and he sweeps serenely over a deepening channel into an infinite sea.”
—Ralph Waldo Emerson (1803–1882)