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.
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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)
“He that will consider the infinite power, wisdom, and goodness of the Creator of all things, will find reason to think it was not all laid out upon so inconsiderable, mean, and impotent a creature as he will find man to be; who, in all probability, is one of the lowest of all intellectual beings.”
—John Locke (1632–1704)