Sequences

Sequences are fundamental mathematical objects consisting of ordered lists of numbers that follow a specific rule or pattern. Sequences are critical in various mathematical concepts, including calculus, series, and number theory. They can model real-world phenomena such as population growth, financial investments, and physical processes like the diminishing height of a bouncing ball.

Each number in a sequence is referred to as a term. Typically, the terms are denoted as a₁, a₂, a₃,…, where the subscript indicates the position within the sequence. When the pattern is evident, sequences often include dots to signify continuation.

Mathematically, a sequence can also be seen as a function whose domain is the set of natural numbers, with each natural number mapped to a specific term. This functional representation is useful when defining sequences explicitly or recursively.

Recursive sequences define each term based on previous terms. One of the most famous examples is the Fibonacci sequence, where each term equals the sum of the two preceding terms:

This definition emphasizes the dependence of each term on its predecessors, which is characteristic of recursive sequences.

Partial sums are the sums of the first few terms of a sequence and are valuable for analyzing how the cumulative sum evolves as more terms are added. For a sequence {a_n}, the n^th partial sum is given by:

Understanding sequences, recursive definitions, and partial sums forms the basis for exploring more advanced topics such as infinite series, convergence, and mathematical induction.

Further, a telescoping sequence is a special type of sequence in which most terms cancel out when the partial sum is expanded, making it easier to evaluate the sum. In a telescoping series, the nth partial sum often reduces to the difference between a few terms, typically the first and the last, allowing for a simplified closed-form expression. This property is especially useful in evaluating infinite series and proving convergence.