Sequence Analysis and Computation

📋 Course Outline

1. Sequence Definition & Elements
2. Explicit Formula & Term Calculation
3. Recurrence Relation & Term Generation
4. Graphical Representation & Visualization
5. Monotonicity & Sequence Behavior
6. Growth & Decay Patterns
7. Sequence Convergence & Limits
8. Algorithmic Computation & Implementation

📖 1. Sequence Definition & Elements

🔑 Key Concepts & Definitions

• Sequence (𝑢ₙ): An ordered list of numbers where each element is associated with an integer index (or rank) 𝑛. Formally, 𝑢ₙ is a function from natural numbers ℕ to real numbers ℝ, assigning each 𝑛 a real term.

• Term of a Sequence: The value associated with a specific position 𝑛 in the sequence, denoted as 𝑢ₙ.

• Index (𝑛): The position or rank of a term in the sequence, typically a natural number starting from 0 or 1.

• Explicit Definition: A formula that directly computes the 𝑛-th term, e.g., 𝑢ₙ = 2𝑛 or 𝑣ₙ = 3𝑛² - 1.

• Recurrence Relation: An equation expressing each term as a function of previous terms, e.g., 𝑢ₙ+1 = 3𝑢ₙ, where the next term depends on the current one.

📝 Essential Points

• A sequence is a list of numbers ordered by a specific rule, which can be explicit (direct formula) or recursive (relation involving previous terms).

• The notation 𝑢ₙ often corresponds to a function 𝑢: ℕ → ℝ, with 𝑢(𝑛) = 𝑢ₙ.

• Computing terms can be done directly using explicit formulas or iteratively via recurrence relations.