Unit 10: Infinite Sequences & Series
Sequences & Convergence
Determining whether a sequence converges or diverges.
Advanced25 min lesson2 min readUpdated August 12, 2026Author not yet attributed
What an Infinite List of Numbers Settles Toward
Definition — Sequence Convergence
A sequence aₙ converges to L if aₙ gets arbitrarily close to L as n → ∞; otherwise it diverges. When aₙ = f(n) for a function f with a known limit as x → ∞, the sequence's limit matches the function's limit. A sequence that is both monotonic (always increasing or always decreasing) and bounded is guaranteed to converge.
Worked Example — Confirming Convergence via a Function Limit
Does aₙ = (2n+1)/(n+3) converge? Treating it as f(x) = (2x+1)/(x+3): dividing numerator and denominator by x gives (2+1/x)/(1+3/x) → 2/1 = 2 as x → ∞. Check numerically: a₁ = 0.75, a₁₀ ≈ 1.615, a₁₀₀ ≈ 1.951, a₁₀₀₀ ≈ 1.995 — approaching 2.
Worked Example — A Sequence That Diverges by Oscillation
Does aₙ = (−1)ⁿ converge? It alternates forever between −1 and 1, never settling near a single value — it diverges, not because it grows unbounded, but because it oscillates.
Tip
A bounded, monotonic sequence can't overshoot a ceiling (or floor) it never crosses — that intuition is exactly why it's guaranteed to converge, without needing to know the limiting value in advance.
Common Mistakes
Assuming every bounded sequence converges, without also checking monotonicity.
(−1)ⁿ is bounded (between −1 and 1) but not monotonic, and it diverges — boundedness alone isn't sufficient.
Key Takeaways
- A sequence's limit, when aₙ = f(n), matches the corresponding function's limit as x → ∞.
- Divergence can happen by growing unbounded or by oscillating without settling.
- A monotonic, bounded sequence is guaranteed to converge.
Summary
The next lesson builds from finite partial sums toward infinite series, and develops tests for determining whether one converges.
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