Sequences and Convergence

Core Idea

A sequence {a_n} converges if lim(n->infinity) a_n = L for some finite L; otherwise it diverges. Convergence of sequences is analyzed using limit laws, the squeeze theorem, the monotone convergence theorem (bounded and monotone implies convergent), and L'Hopital's rule (by treating n as a continuous variable). Sequence convergence is prerequisite to understanding series convergence, since a series converges only if its partial sums form a convergent sequence.

How It's Best Learned

Evaluate limits of sequences algebraically (divide by highest power of n), using L'Hopital's rule, and using the squeeze theorem. Determine monotonicity and boundedness. Practice with geometric sequences (rⁿ), factorial-based sequences (n!/nⁿ), and sequences involving exponentials.

Common Misconceptions

• Confusing sequence convergence with series convergence (the sequence {1/n} converges to 0, but the series sum of 1/n diverges).
• Believing a bounded sequence must converge (it must also be monotone, or use a subsequence argument).
• Not recognizing that L'Hopital's rule applies to sequences by treating a_n = f(n).

Explainer

A sequence is an ordered list of numbers a₁, a₂, a₃, … generated by a rule, such as aₙ = 1/n or aₙ = (−1)ⁿ. Convergence asks a single question: do the terms settle toward a fixed value as n grows without bound? Formally, the sequence converges to L if, for any desired precision ε > 0, there is an index N beyond which every term aₙ is within ε of L. Informally, the terms eventually cluster arbitrarily close to L and never stray away again.

To evaluate convergence, compute lim(n→∞) aₙ using the full toolkit from limits at infinity. For rational-expression sequences, divide numerator and denominator by the highest power of n. For sequences involving exponentials or factorials, L'Hôpital's rule often helps — treat n as a continuous variable x, evaluate lim(x→∞) f(x), and the result applies to the sequence. The squeeze theorem handles cases where the sequence is sandwiched: if bₙ ≤ aₙ ≤ cₙ for all n and both bₙ → L and cₙ → L, then aₙ → L as well.

The monotone convergence theorem provides a qualitative shortcut: a sequence that is simultaneously bounded (all terms between some fixed M and N) and monotone (terms never decrease, or never increase) must converge — even if you cannot find the limit analytically. The classic example is aₙ = (1 + 1/n)ⁿ: it is increasing and bounded above by 3, so it converges. (Its limit is e.) Boundedness alone is not enough — the sequence {(−1)ⁿ} is bounded between −1 and 1 but oscillates forever without converging.

One of the most important distinctions in this course is between sequence convergence and series convergence. The sequence {1/n} converges to 0 — its terms shrink toward nothing. But the series Σ(1/n), the harmonic series, diverges — the running total grows without bound, just very slowly. These are entirely different statements about entirely different objects. A convergent sequence is a prerequisite for a convergent series (if aₙ does not go to 0, the series Σaₙ must diverge), but it is not sufficient.

Sequence convergence is the conceptual foundation for series, improper integrals, and ultimately for the rigorous definition of the real numbers. Building a clear mental model now — terms approaching a limit vs. not — will pay dividends throughout the rest of calculus and analysis.