Sequences for Calculus 2 (Section 11.1)

Definition of a Sequence

• A sequence is a function whose domain is the set of natural numbers (1, 2, 3, ...).
  • Debate on whether 0 is included.
• Represented by aₙ (subscript denotes the term number).
• Function of n: a(n).
• Graph: Consists of isolated points (e.g., (1, a₁), (2, a₂), ...).
  • Can be finite or infinite.
• Example: Sequence term plotting on the n-axis and aₙ axis.

Examples

Example 1: Geometric Sequence

• Given terms: a₁, a₂, a₃ (pattern: multiply by 1/3).
• Common ratio: 1/3.
• General term formula: aₙ = (1/3)^(n-1).
• Verification with substitution.

Example 2: Oscillating Sequence

• Given pattern: 5, 1, 5, 1, ... (oscillates between 5 and 1).
• General term formula: aₙ = 3 + (-1)^(n-1) × 2.
• Verification with substitution.
• Modification: Changing oscillation pattern by raising (-1) to nth power.

Convergent and Divergent Sequences

Definitions

• Convergent Sequence: Limit as n approaches infinity exists and is finite.
• Divergent Sequence: Limit does not exist or equals ±∞.
• Limit Laws for Convergent Sequences (addition, subtraction, multiplication, division, constants).

Theorem: Limit of Composition

• If the limit of aₙ = L and continuous function f, then the limit of f(aₙ) = f(L).

Example 3: Determining Convergence

• Sequence: nth root of 2^(1+3n).
• Simplified and used exponential function's continuity.
• Result: Sequence converges to 8.

Theorem: Absolute Value

• If the limit as n approaches infinity of |aₙ| = 0, then limit of aₙ = 0.

Example 4: Oscillating Sequence

• Sequence: (-1)ⁿ/n.
• Found limit using absolute value and properties of oscillation.
• Result: Sequence converges to 0.
• Visual graph shows oscillation approaching zero.

Squeeze Theorem for Sequences

• Similar to Squeeze Theorem for functions.
• If aₙ ≤ bₙ ≤ cₙ and limit of aₙ and cₙ both equal L, then the limit of bₙ equals L.

Example with Sine Function

• Sequence: sin(2n)/(1 + sqrt(n)).
• Bounded between -1/(1 + sqrt(n)) and 1/(1 + sqrt(n)).
• Limits of bounding sequences both equal 0.
• Conclusion: Original sequence converges to 0 by the squeeze theorem.

Sequence Limit Theorem

• If the limit as x approaches infinity of f(x) = L and f(n) = aₙ for natural numbers n, then limit of aₙ = L.

Example: Applying L'Hôpital's Rule

• Sequence: ln(n²)/n.
• Defined continuous function f(x) = ln(x²)/x.
• Applied L'Hôpital's Rule twice.
• Result: Sequence converges to 0.

Fact on Geometric Sequences

• If |R| < 1, then limit as n approaches infinity of Rⁿ = 0.
• Result: Sequences with common ratios between -1 and 1 shrink to 0.

Example: Geometric Sequence

• Sequence: 3ⁿ + 2 / 5ⁿ.
• Found common ratio and simplified using laws of exponents.
• Result: Sequence converges to 0.

Summary and Definitions

• Increasing Sequence: a₁ < a₂ < a₃ < ...
• Decreasing Sequence: a₁ > a₂ > a₃ > ...
• Monotonic Sequence: Always increasing or always decreasing.
• Bounded Sequence: Has both upper and lower bounds.
• Theorem: Every bounded monotonic sequence converges.

Example: Monotonic and Bounded Sequence

• Sequence: 1/n.
• Bounded above by 1 and below by 0.
• Monotonic (strictly decreasing).
• Conclusion: Sequence converges to 0.

Example: Non-Monotonic Sequence

• Sequence: (-1)^(n+1).
• Bounded above by 1 and below by -1.
• Not monotonic (switches between increasing and decreasing).
• Conclusion: Sequence diverges.

Using Derivatives for Sequence Behavior

• To determine increasing or decreasing, consider a function f(x) analogous to sequence.

Example: Derivative Analysis

• Sequence: (2n-3)/(3n+4).
• Defined function f(x) = (2x-3)/(3x+4).
• Calculated first derivative (positive for all x).
• Conclusion: Sequence is strictly increasing.
• Next consider if bounded (calculate limits).
• Result: Upper bound is 2/3.
• Conclusion: Sequence is bounded and convergent.

Key Points

• Use geometric sequence properties for convergence/divergence checks.
• Apply squeeze theorem to sequences involving sine/cosine.
• Use L'Hôpital's Rule with caution (redefine sequence as function).
• Check for monotonicity and boundedness for convergence.
• Be thorough in showing steps and justifications in proofs.

Stay tuned for section 11.2!