divergéns

Divergence is a fundamental concept in mathematics and physics referring to the property of a sequence, series, or integral that does not converge to a finite value. In the context of sequences, a sequence is said to diverge if its terms do not approach a single, specific number as the index increases indefinitely. This can happen if the terms grow infinitely large, oscillate without settling, or behave in other non-convergent ways.

For infinite series, divergence means that the sum of the terms does not approach a finite limit. This is a common occurrence and tests for series divergence, such as the divergence test, are crucial in calculus. If the terms of a series do not approach zero, the series must diverge. However, the converse is not true; even if the terms approach zero, the series can still diverge.

In vector calculus, divergence is a scalar quantity that measures the rate at which a vector field flows outward from a point. It is a differential operator that takes a vector field and returns a scalar field. A positive divergence at a point indicates a source or outflow of the vector field, while a negative divergence indicates a sink or inflow. A divergence of zero suggests that the field is neither accumulating nor depleting at that point, a property known as being solenoidal. The divergence theorem relates the flux of a vector field through a closed surface to the volume integral of its divergence within the enclosed volume.