Sequence - Doubly Infinite Sequences

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)

    No person can be considered as possessing a good education without religion. A good education is that which prepares us for our future sphere of action and makes us contented with that situation in life in which God, in his infinite mercy, has seen fit to place us, to be perfectly resigned to our lot in life, whatever it may be.
    Ann Plato (1820–?)