Infinite Sequences

Part 1 of 7 — Definition & Convergence

What Is a Sequence?

An ordered list of numbers: $a_1, a_2, a_3, ldots$ or ${a_n}_{n=1}^{infty}$

Convergence

A sequence ${a_n}$ converges to $L$ if:

$lim_{n o infty} a_n = L$

If no such $L$ exists, the sequence diverges.

Examples

Sequence Convergence 🎯

Key Takeaways — Part 1

1. A sequence is a function on the natural numbers
2. "Converges" means the limit exists and is finite

Infinite Sequences

Part 2 of 7 — Bounded & Monotonic Sequences

Monotone Convergence Theorem

If a sequence is bounded and monotonic, it converges.

• Monotonic increasing: $a_{n+1} geq a_n$ for all $n$
• Monotonic decreasing: $a_{n+1} leq a_n$ for all $n$
• Bounded: there exist $M, m$ such that $m leq a_n leq M$

Squeeze Theorem for Sequences

If $b_n leq a_n leq c_n$ and $lim b_n = lim c_n = L$, then $lim a_n = L$.

Bounded & Monotonic 🎯

Key Takeaways — Part 2

Bounded + Monotonic → Converges. Squeeze Theorem works for sequences too!

Infinite Sequences

Part 3 of 7 — Geometric & Recursive Sequences

Geometric Sequences

$a_n = a_1 cdot r^{n-1}$

Converges to $0$ if $|r| < 1$, diverges if $|r| geq 1$ (except $r = 1$, const).

Recursive Sequences

Defined by a recurrence: $a_{n+1} = f(a_n)$

To find the limit $L$: set $L = f(L)$ and solve.

Geometric/Recursive 🎯

Key Takeaways — Part 3

For recursive sequences: set $L = f(L)$ and solve for $L$.

Infinite Sequences

Part 4 of 7 — Growth Rate Hierarchy

Dominance Hierarchy ($n o infty$)

$ln n ll n^p ll a^n ll n! ll n^n$

(for $p > 0$, $a > 1$)

This means: log grows slowest, then polynomial, exponential, factorial, $n^n$ grows fastest.

Example

lim_{n o infty} rac{n^{100}}{2^n} = 0 because exponential beats any polynomial.

Growth Rates 🎯

Key Takeaways — Part 4

$\ln n \ll n^p \ll a^n \ll n! \ll n^n$

Infinite Sequences

Part 5 of 7 — Sequences vs. Series

Critical Distinction

• Sequence: the list $a_1, a_2, a_3, ldots$ → does $a_n$ approach a limit?
• Series: the sum $sum_{n=1}^{infty} a_n$ → does the sum converge?

Divergence Test Preview

If $lim_{n o infty} a_n eq 0$, then $sum a_n$ diverges.

But $lim a_n = 0$ does NOT guarantee convergence! ($sum 1/n$ diverges.)

Sequences vs Series 🎯

Key Takeaways — Part 5

Sequence convergence and series convergence are different questions!

Infinite Sequences

Part 6 of 7 — Practice Workshop

Mixed Practice 🎯

Workshop Complete!

Infinite Sequences — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Infinite Sequences — Complete! ✅