Definition and Notation
Sequence as a Function
Sequence: ordered list of elements indexed by natural numbers. Formal definition: function a: ℕ → ℝ or ℂ. Each term aₙ corresponds to the image of n.
Notation
General term notation: (aₙ) or {aₙ}, n ∈ ℕ. Example: (a₁, a₂, a₃, …). Subscript indicates position.
Examples
Simple sequences: natural numbers (1,2,3,…), even numbers (2,4,6,…), fractional sequences (1/2, 1/3, 1/4,…).
Types of Sequences
Arithmetic Sequences
Definition: difference between consecutive terms constant. Formula: aₙ = a₁ + (n-1)d.
Geometric Sequences
Definition: ratio between consecutive terms constant. Formula: aₙ = a₁ r^{n-1}.
Other Types
Monotonic sequences, bounded sequences, recursive sequences, and complex-valued sequences.
Limits of Sequences
Concept of Limit
Limit: value approached by terms as n → ∞. Denoted limn→∞aₙ = L.
Limit Existence
Limit exists if terms get arbitrarily close to L beyond some index. Otherwise, limit does not exist.
Examples
Sequence aₙ = 1/n converges to 0. Sequence bₙ = (-1)^n does not have a limit.
Convergence and Divergence
Convergent Sequences
Definition: sequence converges if limit exists and is finite. Notation: aₙ → L.
Divergent Sequences
Definition: sequence diverges if limit does not exist or is infinite.
Oscillatory Behavior
Sequences can oscillate without settling to a limit, e.g., aₙ = (-1)^n.
Monotonic Sequences
Increasing Sequences
Definition: aₙ₊₁ ≥ aₙ for all n. Strictly increasing if inequality is strict.
Decreasing Sequences
Definition: aₙ₊₁ ≤ aₙ for all n. Strictly decreasing if strict.
Monotone Convergence Theorem
Every bounded monotonic sequence converges. Critical in analysis and proofs.
Bounded Sequences
Upper and Lower Bounds
Upper bound: number M such that aₙ ≤ M for all n. Lower bound: number m such that aₙ ≥ m.
Boundedness Definition
Sequence bounded if bounded above and below.
Examples
Sequence aₙ = (-1)^n bounded between -1 and 1. Sequence bₙ = n unbounded.
| Sequence | Bounded? | Bounds |
|---|---|---|
| aₙ = (-1)^n | Yes | -1 ≤ aₙ ≤ 1 |
| bₙ = n | No | Unbounded |
Recursive Sequences
Definition
Terms defined by previous terms via recurrence relation. Initial term(s) required.
Examples
Fibonacci sequence: F₁ = 1, F₂ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂.
Solving Recurrences
Methods: characteristic equations, iteration, generating functions, matrix exponentiation.
Fibonacci sequence:F₁ = 1F₂ = 1Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 3Arithmetic Sequences
General Formula
aₙ = a₁ + (n-1)d, where d is common difference.
Sum of Arithmetic Sequence
Formula: Sₙ = n/2 (2a₁ + (n-1)d).
Properties
Linearity: constant difference, linear growth or decay, unbounded unless d=0.
| Term (n) | Value (aₙ) |
|---|---|
| 1 | 3 |
| 2 | 7 |
| 3 | 11 |
| 4 | 15 |
Geometric Sequences
General Formula
aₙ = a₁ r^{n-1}, where r is common ratio.
Sum of Finite Geometric Sequence
Sₙ = a₁ (1 - r^{n}) / (1 - r), for r ≠ 1.
Sum of Infinite Geometric Sequence
Converges if |r| < 1. Sum: S = a₁ / (1 - r).
Geometric sequence sum:Finite: Sₙ = a₁ (1 - rⁿ) / (1 - r), r ≠ 1Infinite: S = a₁ / (1 - r), |r| < 1Applications of Sequences
Calculus
Limits of sequences foundation for series, convergence tests, continuity, and function approximation.
Mathematical Modeling
Population growth, financial modeling, computer algorithms, and physics phenomena.
Computer Science
Algorithm analysis, recursive algorithms, data structure traversal, and complexity estimation.
Common Problems and Examples
Finding Limits
Calculate limn→∞ of sequences using algebraic manipulation or squeeze theorem.
Identifying Types
Recognize arithmetic or geometric nature by differences or ratios.
Sum Calculations
Apply formulae for arithmetic or geometric sums in problem solving.
Advanced Topics
Subsequences
Definition: sequence extracted by selecting terms indexed by increasing subsequence of ℕ. Used in convergence analysis.
Limit Superior and Limit Inferior
Generalize limits for bounded but oscillating sequences. Define lim sup and lim inf as bounds of subsequential limits.
Cauchy Sequences
Definition: sequence where terms become arbitrarily close. Characterizes completeness in metric spaces.
References
- Rudin, W., Principles of Mathematical Analysis, McGraw-Hill, 3rd ed., 1976, pp. 45-72.
- Apostol, T. M., Mathematical Analysis, Addison-Wesley, 2nd ed., 1974, pp. 90-120.
- Stewart, J., Calculus: Early Transcendentals, Cengage Learning, 8th ed., 2015, pp. 100-135.
- Knopp, K., Theory and Application of Infinite Series, Dover Publications, 1990, pp. 33-60.
- Burden, R. L., Faires, J. D., Numerical Analysis, Brooks Cole, 9th ed., 2010, pp. 50-75.