Absolute Convergence
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute value of the summand is finite. More precisely, a real or complex series is said to converge absolutely if for some real or complex number . Similarly, an improper integral of a function, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if
Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess, yet is broad enough to occur commonly. (A convergent series that is not absolutely convergent is called conditionally convergent.)
Read more about Absolute Convergence: Background, Relation To Convergence, Rearrangements and Unconditional Convergence, Products of Series, Absolute Convergence of Integrals
Famous quotes containing the word absolute:
“All forms of beauty, like all possible phenomena, contain an element of the eternal and an element of the transitory—of the absolute and of the particular. Absolute and eternal beauty does not exist, or rather it is only an abstraction creamed from the general surface of different beauties. The particular element in each manifestation comes from the emotions: and just as we have our own particular emotions, so we have our own beauty.”
—Charles Baudelaire (1821–1867)