Introduction to Sequences and Series: Fundamentals, Types, and Applications

Sequences and series are essential mathematical tools. A sequence is an ordered list of numbers, often described by a rule or formula, where each number is a term. Examples include arithmetic sequences (with a constant difference between terms) and geometric sequences (with a constant ratio). In contrast, a series is the sum of terms in a sequence, such as the arithmetic series or geometric series. Sequences can be finite or infinite, while series focus on summing terms, which can converge (approach a specific value) or diverge (grow without bounds). Convergence and divergence are crucial in understanding the behavior of series, especially infinite ones, where convergent series settle toward a limit and divergent series do not. Different types include harmonic, power, and telescoping series, each with distinct properties and applications. Sequences and series are widely used in fields like finance, physics, and data analysis to model patterns, predict behaviors, and evaluate complex summations.

Here are a few examples to illustrate sequences and series:

1. Arithmetic Sequence: \(2, 5, 8, 11, \ldots\), where each term increases by a constant difference of \(3\).
2. Arithmetic Series: Summing the terms of an arithmetic sequence, like \(2 + 5 + 8 + 11 + \ldots\). Its sum can be found using \(\frac{n}{2} \times (\text{first term} + \text{last term})\).
3. Geometric Sequence: \(3, 6, 12, 24, \ldots\), where each term is multiplied by a constant ratio of \(2\).
4. Geometric Series: Adding terms of a geometric sequence, such as \(3 + 6 + 12 + 24 + \ldots\), where the sum formula is \(\frac{\text{first term} \times (1 – \text{ratio}^n)}{1 – \text{ratio}}\) for a finite series, or if the ratio is less than 1, an infinite series converges to \(\frac{\text{first term}}{1 – \text{ratio}}\).
5. Convergence Example: The infinite geometric series \(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots\) converges to \(2\).
6. Divergence Example: The harmonic series \(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots\) diverges, growing infinitely without a finite sum.