Understanding Sequences and Their Limits: Definitions, Examples, and Operations

1 Sequence

A sequence is an ordered list of numbers.

1.1 Example 1

If we have a sequence, the n‑th element is the element at position n. The following figure illustrates an example sequence:

2 Sequence Limits

A sequence has a limit when its terms approach a constant as the index grows indefinitely. This means that for any small tolerance we can find a sufficiently large index such that the difference between the term and the constant is less than that tolerance.

The constant is called the limit of the sequence if and only if for every positive ε there exists an N such that for all n ≥ N, |a_n – L| < ε.

2.1 Example 2

Consider the sequence … The intuitive observation is that its limit is 0; as the index increases, the terms become smaller but never negative.

We now prove this from the definition. To show that for any ε > 0 there exists a positive integer N such that for all n ≥ N the difference between a_n and the limit is less than ε.

The figure below shows how the floor function behaves as the index changes.

2.2 Arithmetic Operations on Limits

If two sequences have limits, the sum, difference, product, and quotient of the sequences have limits equal to the corresponding operations on the individual limits.

3 Summary

This article briefly introduced the definition of sequences and their limits, illustrated the concepts with two examples, and presented the basic arithmetic rules for limits.