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

Graphical or geometric meaning

Points $(n,a_n)$ approach the horizontal line $y=2$. An $\varepsilon$-band around two eventually contains every point, not just infinitely many selected points.

Common mistakes and why they fail

Verification and reasonableness checks

• Compare dominant-term prediction with large-index values.
• Test subsequences when oscillation is suspected.
• Verify an explicit formula satisfies its recurrence and initial condition.

Limits describe eventual behavior

The statement $a_n\to L$ means that every tolerance $\varepsilon>0$ contains all sufficiently late terms within $\varepsilon$ of $L$. Early terms are irrelevant, but persistent oscillation is not. Algebraic limit laws require the component limits and a nonzero limiting denominator. For rational expressions, divide by the highest power of $n$. Every subsequence of a convergent sequence shares the same limit, so two subsequences with different limits prove divergence. Boundedness alone does not guarantee convergence, as $(-1)^n$ shows. A numerical table can suggest a limit but cannot prove one; identify a limit law, squeeze, monotone convergence argument, or direct estimate that controls all sufficiently large indices.

If a formula contains alternating signs multiplied by a factor tending to zero, bound its absolute value and use the squeeze theorem. This separates harmless shrinking oscillation from persistent oscillation whose magnitude never decays.

Practice

1. Find $\lim (3n+1)/(n-2)$.
2. Does $(-1)^n$ converge?
3. Find $\lim(1+1/n)^n$.

Connections and next steps

• Monotone Sequences
• Infinite Series
• Limits at Infinity