Divergents

Divergents is a term used across mathematics to refer to objects that do not converge under a specified limiting process. The precise meaning depends on context. In analysis, a divergent object is one for which a limit does not exist or is unbounded when a procedure such as taking partial sums, limits along a sequence, or integration is applied.

Divergent series and sequences are common contexts. A series ∑ a_n is divergent if the sequence of partial sums S_N = ∑_{n=1}^N a_n does not converge to a finite value. The harmonic series 1 + 1/2 + 1/3 + … diverges. Some series may fail to converge in more subtle ways, or diverge to infinity. In some mathematical theories, divergent series can be assigned finite values through regularization, but such values are not universally regarded as true sums.

A sequence x_n in a metric space diverges if it fails to converge to any point in that space. Divergence can occur when the sequence has no limit, or when it tends to infinity in an extended sense. In complex analysis or real analysis, divergence is distinguished from convergence by the nonexistence of a limit.

Divergence also appears in vector calculus. The divergence of a vector field F = (F1, F2, F3) is the scalar function div F = ∂F1/∂x + ∂F2/∂y + ∂F3/∂z. This quantity measures the rate at which “stuff” is expanding or contracting at a point and is interpreted as a source or sink; fields with zero divergence are called divergence-free.

In addition, improper integrals can be divergent if their limit does not exist, and numerical methods can fail to converge, producing divergent behavior.