# 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.

## Types of Sequences

• Arithmetic Sequences: These sequences have a common difference between consecutive terms. If `a` is the first term and `d` is the common difference, the `n`th 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 `n`th 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.

## Key Properties of Sequences

• 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.

## Summation of Series

• 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)`

## Practical Applications of Sequences and Series

• 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.