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:
“A man calumniated is doubly injured—first by him who utters the calumny, and then by him who believes it.”
—Herodotus (c. 484–425 B.C.)
“Hath not the morning dawned with added light?
And shall not evening call another star
Out of the infinite regions of the night,
To mark this day in Heaven? At last, we are
A nation among nations; and the world
Shall soon behold in many a distant port
Another flag unfurled!”
—Henry Timrod (1828–1867)