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Convergence

Convergence can be defined for infinite sequences of numbers, **functions**, or more general objects such as points in topological spaces. The simplest case to consider is a sequence of numbers, such as (0.9, 0.99, 0.999, 0.9999,...). Such a sequence converges if the terms of the sequence approach a fixed number *L*, called the *limit* of the sequence, to any desired degree of accuracy. The sequence above, for example, converges to the number 1.

A sequence that does not converge is said to diverge. This can happen in various ways: the terms of the sequence can oscillate (1, 0, 1, 0,...) or they can increase without bound (1, 2, 4, 8, 16,...). In the latter case the limit is sometimes said to be infinite, but the sequence is still considered to be divergent.

Convergence can also be defined for sequences of functions (*f*_{1}, *f*_{2}, *f*_{3},...). Such a sequence has a limit function *f* if, for any fixed value of *x*, the limit of...

This section contains 507 words(approx. 2 pages at 300 words per page) |