Search results
Results From The WOW.Com Content Network
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
A series is convergent (or converges) if and only if the sequence (,,, … ) {\displaystyle (S_{1},S_{2},S_{3},\dots )} of its partial sums tends to a limit ; that means that, when adding one a k {\displaystyle a_{k}} after the other in the order given by the indices , one gets partial sums that become closer and closer to a given number.
Calculus ′ = () ... is a strictly monotone and divergent sequence and the following limit exists: ... is finitely convergent if its ratio is less than one ...
The addition of two divergent series may yield a convergent series: for instance, the addition of a divergent series with a series of its terms times will yield a series of all zeros that converges to zero. However, for any two series where one converges and the other diverges, the result of their addition diverges.
A series is said to be convergent if the sequence consisting of its partial sums, (), is convergent; otherwise it is divergent. The sum of a convergent series is defined as the number s = lim n → ∞ s n {\textstyle s=\lim _{n\to \infty }s_{n}} .
If such a limit exists and is finite, the sequence is called convergent. [2] A sequence that does not converge is said to be divergent . [ 3 ] The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero.
Because it is a divergent series, it should be interpreted as a formal sum, an abstract mathematical expression combining the unit fractions, rather than as something that can be evaluated to a numeric value. There are many different proofs of the divergence of the harmonic series, surveyed in a 2006 paper by S. J. Kifowit and T. A. Stamps. [13]