Limit of a sequence

The limit of a sequence is the value its terms approach as their indices grow. A sequence converges when such a value exists.

Epsilon–N definition: A formal definition of convergence requiring every positive tolerance to contain all terms beyond some index. It expresses closeness to the limit using a tolerance and a cutoff index.
Sequence: An ordered list of terms indexed by natural numbers or another specified index set. A limit describes the long-index behavior of this ordered list.
Infinite series: A sum of infinitely many terms defined through the limit of its finite partial sums. A series converges precisely when its sequence of partial sums has a finite limit.
Divergent sequence: A sequence that does not converge to a finite value. It marks the failure of a finite sequence limit to exist.
Convergent sequence: A sequence whose terms approach a finite value as the index grows. Convergence is exactly the condition that gives a sequence a limit.
Absolute value: A real number's distance from zero, extended to complex numbers as modulus. Distance from a proposed limit is measured by the absolute value of the difference.
Taylor series: A power series built from a function's derivatives at a point to represent or approximate that function. Its value is defined by limits of finite polynomial sums where the series converges.
Limit superior: The greatest subsequential limit, generalized to extended real values when necessary. It can describe persistent upper behavior even when no single sequence limit exists.
Cauchy sequence: A sequence whose terms eventually become arbitrarily close to one another. In the real numbers, this internal closeness guarantees a limit without naming it first.
Natural number: A member of the counting numbers, used to index terms of many sequences. The cutoff in the convergence definition is an index chosen among the natural numbers.

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