Sequences of Real Numbers: Convergence and Key Theorems

A sequence of real numbers is an infinite ordered list a₁, a₂, a₃, … — formally, a function from the natural numbers to ℝ. The central question is whether the terms settle down: does aₙ approach some fixed limit L as n grows?

This matters because sequences model every approximation process in mathematics: decimal expansions of √2, iterative root-finding, compound interest over many periods. Convergence is the gateway concept of real analysis.

Key Concepts

• General term: the formula for aₙ, such as aₙ = (2n + 1)/(3n − 2).
• Convergence: aₙ → L means the terms get and stay arbitrarily close to L.
• Limit: the value L the terms approach; a convergent sequence has exactly one limit.
• Divergence: failure to converge — terms may blow up, oscillate, or wander.
• Bounded sequence: all terms lie within some fixed interval; every convergent sequence is bounded.

Formulas

• ε–N definition: lim aₙ = L means for every ε > 0 there exists N such that n > N implies |aₙ − L| < ε.
• Limit laws: limits add, subtract, multiply and divide (denominator non-zero) termwise.
• Squeeze theorem: if aₙ ≤ bₙ ≤ cₙ and aₙ, cₙ → L, then bₙ → L.
• Growth comparison: 1/n → 0, and rⁿ → 0 whenever |r| < 1.

Worked Example

Consider aₙ = (2n + 1)/(3n − 2). Divide numerator and denominator by n: aₙ = (2 + 1/n)/(3 − 2/n). Since 1/n → 0, the limit should be 2/3. Check numerically: n = 10 gives 21/28 = 0.75; n = 100 gives 201/298 ≈ 0.6745; n = 1000 gives 2001/2998 ≈ 0.667556 — visibly closing in on 0.6667. Prove it with ε = 0.01: |aₙ − 2/3| = |(2n+1)/(3n−2) − 2/3| = |7/(3(3n−2))| = 7/(9n − 6). We need 7/(9n − 6) < 0.01, i.e. 9n − 6 > 700, i.e. n > 78.44. So N = 79 works: for every n > 79, the term is within 0.01 of 2/3. Hence the sequence converges to 2/3.

Common Misconceptions

• A sequence must have a formula. Not necessarily — sequences can be defined recursively or as decimal approximations.
• Terms getting closer together implies convergence. In ℚ this fails: a Cauchy sequence of rationals can aim at √2, which is missing — convergence needs the completeness of ℝ.
• A sequence cannot converge to a term it never equals. It can — 1/n converges to 0 without any term being 0.

Real-World Applications

• Numerical root-finding (Newton's method) generates convergent sequences of approximations.
• Compound interest and loan balances follow sequences converging to steady states.
• Computer graphics refine curves through convergent subdivision sequences.

Sequences turn "getting closer and closer" into precise mathematics. Master the ε–N definition on one sequence and every limit argument in analysis follows the same pattern.