Convergence of Sequences

A sequence is a function defined on the set of natural numbers, producing a list of values indexed by n.

If these values approach a single real number as n increases, the sequence converges.

This real number is called the limit of the sequence.

Graphically, this means the values of the sequence get closer to a horizontal line as n increases.

For example, the sequence one over n converges to zero as n approaches infinity.

Two useful tools for proving convergence are boundedness and monotonicity.

A bounded sequence remains within fixed upper and lower values, while a monotonic sequence always increases or decreases.

Together, these lead to the Monotonic Sequence Theorem, which states that if a sequence is both bounded and monotonic, it must converge.

Consider a cooling cup of coffee. Its temperature, measured each minute, forms a decreasing, or monotonic sequence. Since the temperature cannot fall below room temperature, the sequence is bounded below.

Because the sequence is both monotonic and bounded, it converges to the stable room temperature, according to the Monotonic Sequence Theorem.