Understanding Sequences and Series

📋 Course Outline

1. Sequence Definitions
2. Series Notation
3. Arithmetic Sequences
4. Geometric Sequences
5. Harmonic Sequences
6. Recursive Sequences
7. Finite Series
8. Infinite Series
9. Sequence Limits
10. Series Convergence
11. Convergence Tests
12. Harmonic Series Divergence

📖 1. Sequence Definitions

🔑 Key Concepts & Definitions

• Sequence: An ordered list of numbers ( a_n ), where each term corresponds to a specific position ( n ) (usually ( n \in \mathbb{N} )). It can be finite or infinite.
• Term: An individual element in a sequence, denoted as ( a_n ).
• Limit of a Sequence: The value ( L ) that ( a_n ) approaches as ( n \to \infty ), if such a limit exists.
• Convergence: A sequence converges if its terms approach a finite limit ( L ) as ( n ) increases indefinitely.
• Divergence: When a sequence does not approach a finite limit, it diverges (either tends to infinity, negative infinity, or oscillates).