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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.
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.
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.
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. [1]
In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known.
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. [1] Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability.
Calculus ′ = () ... convergence tests are methods of testing for the convergence, ... is a strictly monotone and divergent sequence and the following limit exists:
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. Thus any series in which the individual terms do not approach zero diverges.