Sequences and Series: Properties of sequences

Understanding the Concepts of Sequences and Series

A sequence is a ordered set of numbers where each number is defined as a term. These terms, often denoted as an, are usually generated according to a rule or formula.
A series is the sum of the terms of a sequence. If a sequence is defined by its terms {a1, a2, a3, a4, ...}, the corresponding series is the summation (a1 + a2 + a3 + a4 + ...).
Finite sequences and finite series are those which have a defined number of terms or summands.
Infinite sequences and Infinite series continue indefinitely without an ending.

Arithmetic Sequences: These sequences have a common difference between consecutive terms. If a is the first term and d is the common difference, the nth term is given as an = a + (n-1) * d.
Geometric Sequences: In these sequences, each term after the first is obtained by multiplying the preceding element by a constant, called the common ratio. The nth term is given as an = a * r^(n-1), where a is the first term and r is the common ratio.
Convergent Sequences: A sequence is said to be convergent if it approaches a specific value, known as the limit, as n tends to infinity.
Divergent Sequences: A sequence is said to be divergent if it does not have any finite limit.

Bounded Sequences: A sequence is called bounded if there exists a real number R such that all terms an are less than or equal to R. It’s said to be bounded above if there is a number R such that all an≤R and bounded below if all an≥R.
Monotonic Sequences: A sequence is said to be monotonic if each term is either less than, equal to, or greater than the previous term. If it is increasing, the sequence is called strictly increasing, if it is decreasing, it is called strictly decreasing.

Arithmetic Series: The sum Sn of the first n terms of an arithmetic sequence can be calculated by the formula Sn = n/2 [2a + (n-1) * d].
Geometric Series: The sum Sn of the first n terms of a geometric sequence can be calculated by the formula Sn = a * (1 - r^n) / (1 - r)

Understanding sequences and series plays a critical role in various scientific fields such as physics, engineering, computer science and finance, allowing for the modeling of different types of quantified phenomena.