Understanding Sequences and Their Limits: Key Concepts and Examples

1 Sequence

A sequence is an ordered list of numbers.

1.1 Example 1

If we have a sequence, the nth element is the nth term. For the following sequences:

2 Sequence Limits

A sequence has a limit when its terms approach a constant as the index increases; for any small difference we can find a sufficiently large index such that the term's distance from the constant is less than that difference.

The constant is called the limit of the sequence if and only if for every positive number there exists an index such that: For all subsequent indices the inequality holds.

In this case we write:

2.1 Example 2

Consider the following sequence.

Intuitively we can see its limit is 0. As the index grows larger, the terms become smaller but never negative.

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

Assume we take a value; we can find a suitable N such that the required inequality holds.

For any ε we can always use the following method to find an appropriate index.

Here the floor function is used. The figure below illustrates the size of the terms for different indices.

2.2 Arithmetic of Limits

We can perform the following operations on sequences:

If two sequences have limits, then the sum, difference, product, and quotient of the sequences have limits equal to the sum, difference, product, and quotient of 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.