chapter3

CHAPTER 3: SEQUENCES OF REAL NUMBERS

Contents

• 1. Bounded sequences

• 2. The Convergent Sequences

• 3. Limit Theorems

• 4. Monotonous Sequences

• 5. Particular Sequence

• 6. Extracted Sequences (Subsequences)

• 7. Adjacent Sequences

• 8. The Cauchy Sequences

• 9. Recurring Sequences

1. Bounded Sequences

• Definition 1.1: A sequence is a real-valued function f where the domain is the positive integers (N).

  • Terms: f(1), f(2), ... are called terms of the sequence.

  • Notation: U(n) written as U_n for simplicity.

• Ways to Specify a Sequence:

  1. By giving the function:

     • Example:

       • U_n = 1/n → Sequence: {1, 1/2, 1/3, ...}

       • U_n = (−1)^n n^2 → Sequence: {−1, 4, −9, ...}

  2. By providing the first few terms:

     • Pattern recognition can be tricky.

       • Example:

         • {U_n} = {2, 5/2, 10/3, ...}

  3. By a recursion formula:

     • Example:

       • U_1 = 1, U_(n+1) = 1/(n+1) U_n → Sequence: {1, 1/2, 1/6, ...}

• Definition 1.2: For a sequence (U_n):

  1. If M is an upper bound if U_n ≤ M for all n.

  2. If m is a lower bound if m ≤ U_n for all n.

  3. Bounded above and below if both conditions are met.

• Proposition 1.3: A sequence (U_n) is bounded if |U_n| ≤ k for some k > 0 and all n.

• Definition of Monotonous Sequence (1.4):

  • Increasing: U_n ≤ U_(n+1)

  • Strictly increasing: U_n < U_(n+1)

  • Decreasing: U_n ≥ U_(n+1)

  • Strictly decreasing: U_n > U_(n+1)

• Example 1.5:

  • U_n = 1/(n+1) → Decreasing sequence.

  • U_n = 2^n → Increasing sequence.

2. The Convergent Sequences

• Definition 2.1: A sequence (U_n) converges to l if for every 𝜖 > 0, there exists N(𝜖) such that for all n ≥ N(𝜖), |U_n - l| < 𝜖.

  • Notation: lim (n→∞) U_n = l.

  • If it does not converge, it diverges.

• Example 2.2:

  • Proves

  • lim (n→∞) n^2 - n - 1 / (2n^2 - 1) = 1/2.

• Theorem 2.3: Limits of converging sequences are unique.

3. Limit Theorems

• Proposition 3.1: If lim U_n = 0 and V_n is bounded, then lim (U_n V_n) = 0.

• Theorem 3.3: If (U_n) converges to K and (V_n) converges to L, and U_n ≤ V_n for large n, then K ≤ L.

4. Monotonous Sequences

• Theorem 4.1: If (U_n) is increasing and bounded above, it converges.

• Theorem 4.2: If (U_n) is decreasing and bounded below, it converges.

5. Particular Sequence

• An infinite series is an expression a1 + a2 + a3 + ... = ∑(n=1 to ∞) a_n.

• Convergence criteria for series based on properties of a_n.

6. Extracted Sequences (Subsequences)

• Definition 6.1: Let (U_n) be a sequence. A function ϕ: N → N is strictly increasing; (V_n) = U_(ϕ(n)).

• Theorem 6.3: If (U_n) converges to l, then each subsequence also converges to l.

7. Adjacent Sequences

• Definition 7.1: Two sequences (U_n) and (V_n) are adjacent if (U_n) is increasing, (V_n) is decreasing, and lim (V_n - U_n) = 0.

8. The Cauchy Sequences

• Definition 8.1: A sequence (U_n) is a Cauchy sequence if for every 𝜖 > 0, there exists n0 such that |U_p - U_q| < 𝜖 for p, q ≥ n0.

9. Recurring Sequences

• Definition 9.1: A recurring sequence is defined by recurrences involving previous terms.

• Proposition 9.5: If f: A → A is increasing and (U_n) is defined recursively, then (U_n) is monotonous.

• Example 9.6: Investigates properties of a specific recurring sequence.