What does convergence in numerical sequences refer to?

Convergence in numerical sequences refers to the behavior of a sequence where its terms approach a specific, finite value as the sequence progresses. This concept is fundamental in analysis and forms the basis for understanding limits. When a sequence converges, it means that as you consider more and more terms of the sequence, the terms get arbitrarily close to a particular number, called the limit of the sequence.

For example, if you have a sequence defined by the terms ( a_n = \frac{1}{n} ), as ( n ) increases, the terms ( a_n ) get closer to 0. However, in the context of convergence, we focus on the behavior as ( n ) approaches infinity — in this case, the definite value the terms are approaching is 0.

Convergence is an essential concept not only for sequences but also for series and functions, as it helps in determining the stability and reliability of numerical methods and computations used in engineering and mathematics. Understanding this concept aids in predicting how numerical methods will behave as iterations increase, which is crucial in engineering analysis.