🎯⭐ INTERACTIVE LESSON

Infinite Sequences

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Infinite Sequences - Complete Interactive Lesson

Part 1: Definition & Convergence

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:

$lim_{n \to infty} a_n = L$

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

Examples

Sequence	Limit	Converges?
$a_n = 1/n$	$0$	Yes
$a_n = (-1)^n$

Sequence Convergence 🎯

Key Takeaways — Part 1

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

Part 2: Bounded & Monotonic Sequences

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

Part 3: Geometric & Recursive Sequences

Infinite Sequences

Part 3 of 7 — Geometric & Recursive Sequences

Geometric Sequences

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

Part 4: Growth Rate Hierarchy

Infinite Sequences

Part 4 of 7 — Growth Rate Hierarchy

Dominance Hierarchy ( $n \to infty$ )

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

Part 5: Sequences vs. Series

Infinite Sequences

Part 5 of 7 — Sequences vs. Series

Critical Distinction

Sequence: the list $a_1, a_2, a_3, ldots$ → does approach a limit?

Part 6: Practice Workshop

Infinite Sequences

Part 6 of 7 — Practice Workshop

Mixed Practice 🎯

Workshop Complete!

Part 7: Final Assessment

Infinite Sequences — Review

Part 7 of 7 — Final Assessment

Final Assessment 🎯

Infinite Sequences — Complete! ✅

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!

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

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 $L$ .

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

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

$lim_{n \to infty} \frac{n^{100}}{2^n} = 0$ because exponential beats any polynomial.

Growth Rates 🎯

Key Takeaways — Part 4

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

Series: the sum

sum_{n=1}^{infty} a_n

→ does the sum converge?

Divergence Test Preview

If $lim_{n \to infty} a_n \neq 0$ , 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

Infinite Sequences - Complete Interactive Lesson

Part 1: Definition & Convergence

Infinite Sequences

What Is a Sequence?

Convergence

Examples

Key Takeaways — Part 1

Part 2: Bounded & Monotonic Sequences

Infinite Sequences

Monotone Convergence Theorem

Part 3: Geometric & Recursive Sequences

Infinite Sequences

Geometric Sequences

Part 4: Growth Rate Hierarchy

Infinite Sequences

Dominance Hierarchy (n→inftyn \to inftyn→infty)

Part 5: Sequences vs. Series

Infinite Sequences

Critical Distinction

Part 6: Practice Workshop

Infinite Sequences

Workshop Complete!

Part 7: Final Assessment

Infinite Sequences — Review

Infinite Sequences — Complete! ✅

Squeeze Theorem for Sequences

Key Takeaways — Part 2

Recursive Sequences

Key Takeaways — Part 3

Example

Key Takeaways — Part 4

Divergence Test Preview

Key Takeaways — Part 5

Dominance Hierarchy ( $n \to infty$ )