Infinite

In contrast, a sequence that is infinite in both directions—i.e. that has neither a first nor a final element—is called a bi-infinite sequence, two-way infinite sequence, or doubly infinite sequence.

If God is an imperfect Being, submission to his will may be a profound mistake; but if he is infinite in wisdom, infinite in holiness, infinite in love, then submission to his will must be the brightest and noblest end of life.

Graphically, these are the infinite affine hyperplane, the infinite hyper-octant, and the infinite simplex.

History of the theory of infinite series Development of infinite series Greek mathematician Archimedes produced the first known summation of an infinite series with a method that is still used in the area of calculus today.

Rephrased, for any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers.

This shows that it is not possible to use a generic symbol such as ∞ for all the infinite quantities in the hyperreal system; infinite quantities differ in magnitude from other infinite quantities, and infinitesimals from other infinitesimals.

In case you’re not familiar, the infinite monkey theorem provides that a monkey hitting keys at random on a typewriter given an infinite amount of time will almost surely type the complete works of William Shakespeare.

But since you are visiting A an infinite number of times, you must also visit the four squares horizontally and vertically adjacent to A an infinite number of times.

A complete binary tree * In the infinite complete binary tree, every node has two children (and so the set of levels is countably infinite ).

Ampère's law is always valid for steady currents and can be used to calculate the B -field for certain highly symmetric situations such as an infinite wire or an infinite solenoid.

An infinite-dimensional Banach space X is said to be homogeneous if it is isomorphic to all its infinite-dimensional closed subspaces.

Any frequency modulated signal will have an infinite number of sidebands and hence an infinite bandwidth but, in practice, all significant sideband energy (98% or more) is concentrated within the bandwidth defined by Carson's rule.

A seeming paradox, non-standard models of the theory of hereditarily finite sets contain infinite sets --- but these infinite sets look finite from within the model.

As long as the density of stars throughout the universe remains constant, regardless of whether the universe itself has a finite or infinite age, there would be infinitely many other stars in the same angular direction, with an infinite total impact.

Because at least one such infinite schema is required (ZFC is not finitely axiomatizable), this shows that the axiom schema of replacement can stand as the only infinite axiom schema in ZFC if desired.

Chomsky frequently quotes Humboldt's description of language as a system which "makes infinite use of finite means", meaning that an infinite number of sentences can be created using a finite number of grammatical rules.

Comparing the convergents with the approximants derived from the Babylonian method: : : : : : Properties of infinite continued fractions Baire space is a topological space on infinite sequences of natural numbers.

Expressiveness The Löwenheim–Skolem theorem shows that if a first-order theory has any infinite model, then it has infinite models of every cardinality.

For any infinite sets A and B, if there is an injection from A to B then there is an injection from subsets of A to subsets of B. Thus for any infinite cardinals A and B, : If A and B are finite, the stronger inequality : holds.

For example, in the Leibniz formula, an infinite sum (all of whose terms are infinite products) would have to be calculated.