2.3 Convergence and Divergence

Sequences and their limits are crucial in understanding mathematical analysis. Convergent sequences approach a specific value, while divergent sequences don't settle on a particular limit. This distinction is key to grasping the behavior of infinite processes.

Convergence and divergence criteria help determine a sequence's fate. The epsilon-N definition provides a rigorous way to prove convergence, while the Squeeze Theorem offers a handy tool for tricky sequences. These concepts form the foundation for analyzing more complex mathematical structures.

Convergent vs Divergent Sequences

Defining Sequences and Their Behavior

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• A sequence is an ordered list of numbers, denoted as {a_n}, where n is a positive integer
• Sequences can exhibit different behaviors:
  • Converging to a specific limit (convergent sequences)
  • Diverging by approaching infinity, negative infinity, or oscillating without approaching any specific value (divergent sequences)

Convergence and Divergence Criteria

• A sequence {a_n} converges to a limit L if, for any arbitrarily small positive number ε, there exists a positive integer N such that |a_n - L| < ε for all n ≥ N
  • In this case, the sequence is called convergent
  • Example: The sequence {1/n} converges to 0 as n approaches infinity
• A sequence {a_n} diverges if it does not converge to any limit
  • In this case, the sequence is called divergent
  • Examples of divergent sequences:
    • {n} diverges to infinity as n approaches infinity
    • {(-1)^n} oscillates between 1 and -1 without approaching any specific value

Convergence of Sequences

Proving Convergence or Divergence Using the Definition of Limit

• To prove that a sequence {a_n} converges to a limit L using the definition, one must:
  • Find a suitable N for any given ε > 0 such that |a_n - L| < ε for all n ≥ N
  • Example: To prove that {1/n} converges to 0, choose N > 1/ε for any given ε > 0
• To prove that a sequence {a_n} diverges using the definition, one must:
  • Show that for some ε > 0, there does not exist an N such that |a_n - L| < ε for all n ≥ N, regardless of the choice of L
  • Example: To prove that {n} diverges, choose ε = 1 and show that for any N, there exists an n ≥ N such that |n - L| ≥ 1
• The choice of ε and N in the definition of the limit is crucial in determining the convergence or divergence of a sequence

Applying the Definition of Limit to Specific Sequences

• The definition of limit can be applied to various sequences to determine their convergence or divergence
  • Examples of sequences that can be analyzed using the definition of limit:
    • Constant sequences (e.g., {3})
    • Rational sequences (e.g., {1/n}, {n/(n+1)})
    • Exponential sequences (e.g., {2^(-n)}, {(1+1/n)^n})
• Applying the definition of limit may involve algebraic manipulations, inequalities, and the properties of absolute values

The Squeeze Theorem

Statement and Conditions of the Squeeze Theorem

• The Squeeze Theorem (also known as the Sandwich Theorem or Pinching Theorem) states:
  • If {a_n} ≤ {b_n} ≤ {c_n} for all n greater than some N, and both {a_n} and {c_n} converge to the same limit L, then {b_n} also converges to L
• To apply the Squeeze Theorem, one must:
  • Find suitable lower and upper bounding sequences {a_n} and {c_n} that converge to the same limit L
  • Show that the sequence of interest {b_n} is "squeezed" between {a_n} and {c_n}

Applications of the Squeeze Theorem

• The Squeeze Theorem is particularly useful when dealing with sequences involving:
  • Trigonometric functions (e.g., {sin(1/n)})
  • Other oscillating terms (e.g., {(-1)^n/n})
• Examples of applying the Squeeze Theorem:
  • To prove that {sin(1/n)} converges to 0, use the inequalities -1/n ≤ sin(1/n) ≤ 1/n and the fact that both {-1/n} and {1/n} converge to 0
  • To prove that {n sin(1/n)} converges to 1, use the inequalities n(1/n - 1/6n^3) ≤ n sin(1/n) ≤ n(1/n) and the fact that both {n(1/n - 1/6n^3)} and {n(1/n)} converge to 1

Geometric vs Harmonic Sequences

Properties of Geometric Sequences

• A geometric sequence is a sequence of the form {a_n} = {ar^(n-1)}, where:
  • a is the first term
  • r is the common ratio (the ratio between consecutive terms)
• Convergence and divergence of geometric sequences:
  • A geometric sequence converges if |r| < 1
  • A geometric sequence diverges if |r| ≥ 1
• The sum of the first n terms of a geometric sequence is given by:
  • S_n = a(1 - r^n) / (1 - r) for r ≠ 1
  • S_n = na for r = 1

Properties of Harmonic Sequences

• A harmonic sequence is a sequence of the form {a_n} = {1/n}
  • The terms of a harmonic sequence are the reciprocals of the positive integers
• The harmonic sequence diverges, but its series (the sum of the terms) is the harmonic series, which also diverges
  • The divergence of the harmonic series can be proved using the integral test or the comparison test
• The harmonic sequence has applications in various fields, such as:
  • Physics (e.g., the intensity of sound waves)
  • Geometry (e.g., the lengths of the segments in a harmonic division of a line segment)

Other Important Classes of Sequences

• Arithmetic sequences: Sequences with a common difference between consecutive terms
  • Example: {2, 5, 8, 11, 14, ...} (common difference: 3)
• Alternating sequences: Sequences with terms alternating in sign
  • Example: {1, -1/2, 1/3, -1/4, 1/5, ...}
• Monotonic sequences: Sequences that are either non-increasing or non-decreasing
  • Example of a non-increasing sequence: {1/n}
  • Example of a non-decreasing sequence: {1 - 1/n}