Sequences

The central idea

A sequence is a function whose domain is typically the positive integers. We write $a_n$ for its $n$th term and $\lim_{n\to\infty}a_n=L$ when terms become arbitrarily close to $L$ for all sufficiently large $n$. A sequence may be explicit, recursive, monotone, bounded, oscillatory, or divergent to infinity.

Definitions, hypotheses, and notation

Two subsequences with different limits prove the original sequence diverges; for $(-1)^n$, even and odd subsequences approach $1$ and $-1$. Squeeze arguments can prove limits when direct algebra is difficult. Every convergent sequence is bounded, though boundedness alone does not guarantee convergence.

Exponential growth dominates polynomial growth, and factorial growth dominates fixed-base exponentials in common positive examples. These hierarchies guide limit simplification. Indeterminate forms describe competing behavior and call for analysis; they are not final limit values. For recursive formulas, verify both the initial index and every domain restriction before using induction or fixed-point reasoning.

Conceptual meaning

A sequence records discrete evolution. Its limit ignores any finite beginning and describes tail behavior. Graphically the input values are isolated integers, so the sequence is not the same object as a continuous curve drawn through its points.

A dependable method and decision rule

1. List several terms while preserving the exact indexing rule.
2. Identify dominant powers, exponential factors, or recurrence structure.
3. Apply limit laws only when their hypotheses are satisfied.
4. For recurrences, prove convergence before passing a limit through the update equation.
5. State finite limit, infinite divergence, or nonexistence from oscillation.

Fully worked example

Common mistakes and why they fail