2.2 Limit of a Sequence

Sequences and their limits are fundamental concepts in mathematical analysis. They provide a foundation for understanding how mathematical patterns behave as they extend infinitely. By studying sequences, we gain insights into convergence, divergence, and the behavior of functions at extreme values.

Limits of sequences help us determine the long-term behavior of mathematical patterns. We explore techniques for evaluating these limits, including algebraic manipulation and the Squeeze Theorem. Understanding sequence limits is crucial for grasping more advanced calculus concepts and real-world applications.

Limits of Sequences

Understanding Sequences and Their Limits

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• A sequence is a function whose domain is the set of positive integers, typically denoted as {a_n} or (a_n)
• The limit of a sequence is the value that the terms of the sequence approach as the index n approaches infinity, denoted as lim_(n→∞) a_n = L
• If a sequence has a limit, it is said to be convergent, while a sequence without a limit is said to be divergent
• A sequence can converge to a finite value (positive or negative), positive or negative infinity, or oscillate without approaching a specific value
• The behavior of a sequence can be analyzed by examining its terms and the pattern they follow as n increases

Properties and Behavior of Sequences

• Constant sequences, where all terms are equal to a constant value, always converge to that constant value
• Geometric sequences, where each term is a constant multiple of the previous term, can converge or diverge depending on the value of the common ratio
• Sequences involving rational functions, such as polynomials divided by polynomials, can be analyzed by examining the degrees of the numerator and denominator polynomials
• Sequences can exhibit different behaviors, such as monotonically increasing or decreasing, bounded above or below, or alternating between positive and negative values

Convergence vs Divergence

Applying the Definition of the Limit

• The formal definition of the limit of a sequence states that a sequence {a_n} converges to a limit L if, for every ε > 0, there exists a positive integer N such that |a_n - L| < ε for all n ≥ N
• To prove that a sequence converges to a limit using the definition, one must find an appropriate value of N for any given ε > 0
• If a sequence does not satisfy the definition of the limit for any value of L, then the sequence is divergent
• The definition of the limit can be used to prove the convergence or divergence of sequences, such as constant sequences ($a_n = c$), geometric sequences ($a_n = ar^n$), and sequences involving rational functions ($a_n = \frac{P(n)}{Q(n)}$)

Examples of Convergent and Divergent Sequences

• The sequence $a_n = \frac{1}{n}$ converges to 0 as n approaches infinity
• The sequence $a_n = (-1)^n$ diverges as it oscillates between 1 and -1 without approaching a specific value
• The sequence $a_n = \frac{n^2 + 1}{n^2 - 1}$ converges to 1 as the degrees of the numerator and denominator polynomials are equal
• The sequence $a_n = \frac{2n + 3}{n - 1}$ diverges to positive infinity as the degree of the numerator is greater than the degree of the denominator

Evaluating Limits

Algebraic Manipulation Techniques

• Algebraic manipulation can be used to simplify the terms of a sequence and determine its limit, if it exists
• Common algebraic techniques include factoring, rationalization, and applying the properties of limits (sum, difference, product, and quotient rules)
• Factoring can be used to cancel common factors in the numerator and denominator of a rational expression
• Rationalization involves multiplying the numerator and denominator by the conjugate of the denominator to eliminate radicals or complex numbers in the denominator

Other Techniques for Evaluating Limits

• The Squeeze Theorem (or Sandwich Theorem) can be used to determine the limit of a sequence by comparing it with two other sequences that have the same limit
• If a sequence {a_n} satisfies the inequalities {b_n} ≤ {a_n} ≤ {c_n} for all n ≥ N and lim_(n→∞) b_n = lim_(n→∞) c_n = L, then lim_(n→∞) a_n = L
• The limit of a sequence can also be determined using the properties of continuous functions, such as the composition of continuous functions and the limit of a function at infinity
• If f(x) is a continuous function and lim_(n→∞) a_n = L, then lim_(n→∞) f(a_n) = f(L)

Uniqueness of Limits

Proving the Uniqueness of the Limit

• The uniqueness of the limit states that if a sequence converges, then its limit is unique
• To prove the uniqueness of the limit, assume that a sequence {a_n} converges to two different limits, L1 and L2
• Using the definition of the limit, show that for any ε > 0, there exist positive integers N1 and N2 such that |a_n - L1| < ε/2 for all n ≥ N1 and |a_n - L2| < ε/2 for all n ≥ N2
• Choose N = max(N1, N2) and apply the triangle inequality to show that |L1 - L2| < ε for any ε > 0
• Conclude that L1 = L2, proving the uniqueness of the limit by contradiction

Implications of the Uniqueness of Limits

• The uniqueness of limits ensures that a convergent sequence cannot have more than one limit value
• This property is crucial for the consistency and well-definedness of the limit concept in mathematical analysis
• The uniqueness of limits allows for the comparison and manipulation of limits without ambiguity
• It also enables the application of limit properties, such as the sum, difference, product, and quotient rules, to evaluate the limits of more complex sequences