Sequences: Definitions and Examples

Overview

General term a_n: formula producing the nth term for arbitrary n.
Geometric sequence example:
- Terms: 1, 1/3, 1/9, ...
- Formula: a_n = (1/3)^(n-1).
- Check: substitute n = 1 gives 1.
Oscillating two-value sequence example:
- Terms: 5, 1, 5, 1, ...
- Midline method: midpoint = 3; amplitude = 2.
- Formula: a_n = 3 + 2(-1)^(n-1) (gives odd terms 5, even terms 1).
- Variant (swap sign pattern): a_n = 3 + 2(-1)^n.
Geometric with factor extraction:
- Example: (3^(n+2))/(5^n) = 9*(3/5)^n.*

Limit of a sequence: lim_{n→∞} a_n = L.
Convergent sequence: limit exists and is finite.
Divergent sequence: limit does not exist or equals ±∞.
Monotonic:
- Increasing: a_1 < a_2 < a_3 < ...
- Decreasing: a_1 > a_2 > a_3 > ...
- Monotonic sequence: either always increasing or always decreasing.
Bounded:
- Bounded above: exists H s.t. a_n ≤ H for all n.
- Bounded below: exists m s.t. a_n ≥ m for all n.
- Bounded sequence: bounded both above and below._

Limit laws for convergent sequences: same algebraic rules as function limits.
- Sum, difference, product, quotient (denominator not 0), constant multiple, powers.
Composition with continuous function:
- If lim_{n→∞} a_n = L and f is continuous at L, then lim_{n→∞} f(a_n) = f(L).
Absolute value criterion:
- If lim_{n→∞} |a_n| = 0 then lim_{n→∞} a_n = 0.
Squeeze theorem for sequences:
- If a_n ≤ b_n ≤ c_n (for n ≥ N) and lim a_n = lim c_n = L, then lim b_n = L.
Function-sequence relation:
- If f(x) → L as x→∞ and a_n = f(n), then lim_{n→∞} a_n = L.
- Use to apply tools like L'Hôpital by defining f(x) first._

Geometric limit:
- If |r| < 1 then lim_{n→∞} r^n = 0.
- If r = 1 then limit = 1.
Polynomial-ratio sequences:
- For rational expressions in n, compare degrees: leading-coefficient ratio gives limit.
L'Hôpital's rule for sequences:
- Not applied directly to sequences; define f(x) with x real, take limit x→∞, then infer sequence limit.
- Example: a_n = ln(n^2)/n → define f(x) = ln(x^2)/x, apply L'Hôpital twice → limit 0._

Example: lim_{n→∞} (2^(1+3n))^(1/n) = 2^{lim (1+3n)/n} = 2^3 = 8.
- Concludes sequence converges to 8.
Example: a_n = (-1)^n / n.
- Use absolute value: |a_n| = 1/n → 0, so a_n → 0 (oscillatory but decaying).
Example (squeeze): a_n = sin(2n)/(1+√n).
- Bound: -1/(1+√n) ≤ a_n ≤ 1/(1+√n); both bounds → 0, so a_n → 0.
Example: a_n = 3^(n+2)/5^n = 9*(3/5)^n → since |3/5|<1, a_n → 0.
Example using L'Hôpital (sequence → function):
- a_n = ln(n^2)/n; f(x)=ln(x^2)/x; apply L'Hôpital → limit 0; sequence converges.*_

Theorem: Every bounded monotonic sequence converges.
- Need both monotonicity (increasing or decreasing) and boundedness.
- Intuition: monotonic with bounds must approach a finite limit.
Examples:
- a_n = 1/n: bounded (0 ≤ a_n ≤ 1), decreasing → converges to 0.
- a_n = (-1)^{n+1}: bounded (between -1 and 1) but not monotonic → diverges.

To test monotonicity of a sequence given by a rational expression, define f(x) and use derivative:
- Example: a_n = (2n - 3)/(3n + 4).
- Define f(x) = (2x-3)/(3x+4). Compute f'(x) via quotient rule.
- f'(x) = 17/(3x+4)^2 > 0 for all x → function strictly increasing.
- Hence sequence is strictly increasing.
Bounds for that example:
- Lower bound = a_1 = -1/7 (since increasing).
- Upper bound = limiting value as n→∞ = ratio of leading coefficients = 2/3.
- Thus sequence is bounded and convergent (limit 2/3).

Recognize geometric sequences; test |r| < 1 for convergence.
Use squeeze theorem often for sequences with sine/cosine factors.
For rational-type sequences, either divide by highest power or use L'Hôpital after defining f(x).
To test monotonicity, convert to function f(x), take derivative, and infer sequence behavior.
To apply L'Hôpital or derivative tests, always first define an appropriate f(x) with real domain.