🎯 Introduction to Sequences

What is a Sequence?

A sequence is an ordered list of numbers:

$a_1, a_2, a_3, a_4, \ldots, a_n, \ldots$

Each number is called a term.

Notation:

• Individual terms: $a_1, a_2, a_3, \ldots$
• General term: $a_n$ (the nth term)
• Whole sequence: $\{a_n\}$ or $\{a_n\}_{n=1}^{\infty}$

💡 Key Idea: A sequence is a function from positive integers to real numbers! $a_n = f(n)$

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Explicit Formula

A formula that gives $a_n$ directly in terms of $n$.

Example 1: $a_n = \frac{1}{n}$

$a_1 = 1, \quad a_2 = \frac{1}{2}, \quad a_3 = \frac{1}{3}, \quad a_4 = \frac{1}{4}, \ldots$

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Example 2: $a_n = (-1)^n \cdot n$

$a_1 = -1, \quad a_2 = 2, \quad a_3 = -3, \quad a_4 = 4, \ldots$

Alternating signs!

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Recursive Formula

Defines each term using previous term(s).

Example 3: $a_1 = 1$, $a_n = 2a_{n-1}$ for $n \geq 2$

$a_1 = 1$ $a_2 = 2(1) = 2$ $a_3 = 2(2) = 4$ $a_4 = 2(4) = 8$

This is the sequence $1, 2, 4, 8, 16, \ldots$ (powers of 2!)

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Example 4 (Fibonacci): $a_1 = 1$, $a_2 = 1$, $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$

$1, 1, 2, 3, 5, 8, 13, 21, \ldots$

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Finding Patterns

Given the first few terms, find the formula.

Example: $2, 4, 6, 8, 10, \ldots$

Pattern: Even numbers!

$a_n = 2n$

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Example: $1, 4, 9, 16, 25, \ldots$

Pattern: Perfect squares!

$a_n = n^2$

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Example: $1, -1, 1, -1, 1, \ldots$

Pattern: Alternating!

$a_n = (-1)^{n+1}$ (or $(-1)^{n-1}$)

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Limit of a Sequence

The sequence $\{a_n\}$ converges to limit $L$ if:

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

Meaning: As $n$ gets larger, $a_n$ gets closer and closer to $L$.

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If the limit exists (finite): sequence converges

If the limit doesn't exist or is infinite: sequence diverges

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Example 5: Find $\lim_{n \to \infty} \frac{1}{n}$

As $n \to \infty$, the denominator grows without bound.

$\lim_{n \to \infty} \frac{1}{n} = 0$

The sequence converges to 0.

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Example 6: Find $\lim_{n \to \infty} \frac{2n + 1}{3n - 2}$

Method: Divide numerator and denominator by highest power of $n$ (which is $n^1$).

$\lim_{n \to \infty} \frac{2n + 1}{3n - 2} = \lim_{n \to \infty} \frac{\frac{2n}{n} + \frac{1}{n}}{\frac{3n}{n} - \frac{2}{n}}$

$= \lim_{n \to \infty} \frac{2 + \frac{1}{n}}{3 - \frac{2}{n}}$

As $n \to \infty$: $\frac{1}{n} \to 0$ and $\frac{2}{n} \to 0$

$= \frac{2 + 0}{3 - 0} = \frac{2}{3}$

The sequence converges to $\frac{2}{3}$.

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Example 7: Find $\lim_{n \to \infty} (-1)^n$

The sequence is: $-1, 1, -1, 1, -1, 1, \ldots$

It oscillates between -1 and 1, never settling down.

$\lim_{n \to \infty} (-1)^n \text{ does not exist}$

The sequence diverges.

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Limit Laws for Sequences

If $\lim_{n \to \infty} a_n = A$ and $\lim_{n \to \infty} b_n = B$, then:

1. Sum: $\lim_{n \to \infty} (a_n + b_n) = A + B$

2. Difference: $\lim_{n \to \infty} (a_n - b_n) = A - B$

3. Product: $\lim_{n \to \infty} (a_n \cdot b_n) = A \cdot B$

4. Quotient: $\lim_{n \to \infty} \frac{a_n}{b_n} = \frac{A}{B}$ (if $B \neq 0$)

5. Constant Multiple: $\lim_{n \to \infty} c \cdot a_n = c \cdot A$

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Squeeze Theorem for Sequences

If $a_n \leq b_n \leq c_n$ for all $n$, and:

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

Then:

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

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Example 8: Use Squeeze Theorem

Find $\lim_{n \to \infty} \frac{\sin n}{n}$.

Step 1: Recall $-1 \leq \sin n \leq 1$ for all $n$.

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Step 2: Divide by $n$ (positive):

$-\frac{1}{n} \leq \frac{\sin n}{n} \leq \frac{1}{n}$

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Step 3: Take limits:

$\lim_{n \to \infty} -\frac{1}{n} = 0 \quad \text{and} \quad \lim_{n \to \infty} \frac{1}{n} = 0$

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Step 4: By Squeeze Theorem:

$\lim_{n \to \infty} \frac{\sin n}{n} = 0$

The sequence converges to 0.

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Monotonic Sequences

Increasing: $a_n \leq a_{n+1}$ for all $n$

Decreasing: $a_n \geq a_{n+1}$ for all $n$

Monotonic: either increasing or decreasing

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Bounded Sequences

Bounded above: $a_n \leq M$ for some $M$ and all $n$

Bounded below: $a_n \geq m$ for some $m$ and all $n$

Bounded: both bounded above and below

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Monotone Convergence Theorem

If a sequence is monotonic and bounded, then it converges.

💡 Key Idea: If sequence is increasing but can't go past some ceiling, it must level off!

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Example 9: Show $a_n = \frac{n}{n+1}$ converges

Step 1: Find limit

$\lim_{n \to \infty} \frac{n}{n+1} = \lim_{n \to \infty} \frac{1}{1 + \frac{1}{n}} = \frac{1}{1+0} = 1$

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Step 2: Check if increasing

$a_{n+1} = \frac{n+1}{n+2}$

Compare: Is $\frac{n+1}{n+2} \geq \frac{n}{n+1}$?

Cross-multiply: $(n+1)^2 \geq n(n+2)$

$n^2 + 2n + 1 \geq n^2 + 2n$

$1 \geq 0$ ✓

Sequence is increasing.

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Step 3: Check if bounded

$a_n = \frac{n}{n+1} < 1$ for all $n$

Also, $a_n > 0$ for all $n$.

Sequence is bounded: $0 < a_n < 1$

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Conclusion: By Monotone Convergence Theorem, sequence converges (to 1).

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Common Sequence Limits

| Sequence | Limit | |----------|-------| | $\frac{1}{n^p}$ (p > 0) | 0 | | $\frac{1}{n}$ | 0 | | $a^n$ (if abs(a) < 1) | 0 | | $a^n$ (if a > 1) | ∞ | | $\frac{\ln n}{n}$ | 0 | | $n^{1/n}$ | 1 | | $\left(1 + \frac{1}{n}\right)^n$ | e |

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L'Hôpital's Rule for Sequences

If $\lim_{n \to \infty} a_n$ has indeterminate form, treat $n$ as continuous variable $x$ and use L'Hôpital's Rule!

Example: $\lim_{n \to \infty} \frac{\ln n}{n}$

This is $\frac{\infty}{\infty}$ form.

Replace $n$ with $x$:

$\lim_{x \to \infty} \frac{\ln x}{x} = \lim_{x \to \infty} \frac{1/x}{1} = \lim_{x \to \infty} \frac{1}{x} = 0$

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Example 10: $\lim_{n \to \infty} \frac{n^2}{e^n}$

This is $\frac{\infty}{\infty}$ form.

Use L'Hôpital's Rule (twice):

$\lim_{x \to \infty} \frac{x^2}{e^x} = \lim_{x \to \infty} \frac{2x}{e^x}$ (still $\frac{\infty}{\infty}$)

$= \lim_{x \to \infty} \frac{2}{e^x} = 0$

The sequence converges to 0.

Exponentials grow faster than polynomials!

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⚠️ Common Mistakes

Mistake 1: Confusing Sequence Index

$a_1$ is the FIRST term, not $a_0$ (unless stated otherwise).

Be careful with initial conditions!

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Mistake 2: Thinking Convergence Means Limit is 0

WRONG: "If $\{a_n\}$ converges, then $\lim a_n = 0$"

EXAMPLE: $a_n = 1 + \frac{1}{n}$ converges to 1, not 0!

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Mistake 3: Using Wrong Variable

When using L'Hôpital's Rule, replace discrete $n$ with continuous $x$.

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Mistake 4: Ignoring Oscillation

$a_n = \cos(n\pi)$ oscillates: $1, -1, 1, -1, \ldots$

This diverges! Don't assume every formula converges.

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📝 Practice Strategy

1. Find first few terms to see the pattern
2. For limits: divide by highest power, use L'Hôpital's if needed
3. Check for oscillation: look for $(-1)^n$ or trig functions
4. Use Squeeze Theorem for bounded oscillating terms
5. Monotonic + Bounded → converges (even if you can't find exact limit)
6. Common limits: memorize $\frac{1}{n^p} \to 0$, $r^n \to 0$ if $|r| < 1$
7. Exponentials beat polynomials: $e^n$ grows faster than $n^k$