An infinite sequence is an ordered list of numbers: $a_1, a_2, a_3, \dots, a_n, \dots$. We write $\{a_n\}_{n=1}^\infty$ or simply $\{a_n\}$. The study of sequences is the foundation for understanding infinite series — a major topic in BC Calculus.
🎯 In this section you will learn
- What an infinite sequence is and how to write its terms
- How to find the limit of a sequence as $n \to \infty$
- Convergent and divergent sequences
- Monotonic (increasing/decreasing) and bounded sequences
- The Monotonic Sequence Theorem
📌 Definition of a Sequence
$$ \{a_n\}_{n=1}^\infty = a_1, a_2, a_3, \dots $$
A sequence is a function whose domain is the positive integers
Write the first four terms of the sequence $a_n = \frac{n}{n+1}$.
- $a_1 = \frac{1}{2}$
- $a_2 = \frac{2}{3}$
- $a_3 = \frac{3}{4}$
- $a_4 = \frac{4}{5}$
📌 Limit of a Sequence
$$ \lim_{n \to \infty} a_n = L \quad \text{means that } a_n \text{ gets arbitrarily close to } L \text{ as } n \to \infty $$
💡 Convergence vs Divergence
- If $\lim_{n \to \infty} a_n$ exists (as a finite number), the sequence converges.
- If the limit is infinite or does not exist, the sequence diverges.
Determine if $a_n = \frac{1}{n}$ converges or diverges.
- As $n \to \infty$, $\frac{1}{n} \to 0$.
- Since the limit exists (0), the sequence converges to $0$.
Determine if $a_n = n$ converges or diverges.
- As $n \to \infty$, $n \to \infty$.
- Since the limit is infinite, the sequence diverges.
Determine if $a_n = (-1)^n$ converges or diverges.
- Terms: $-1, 1, -1, 1, \dots$
- The terms do not approach a single value. The limit does not exist, so the sequence diverges.
📌 Finding Limits of Sequences
We can often use limit laws and L'Hôpital's rule to find limits of sequences.
Find $\lim_{n \to \infty} \frac{\ln n}{n}$.
- As $n \to \infty$, $\frac{\infty}{\infty}$ form.
- Apply L'Hôpital: $\lim_{n \to \infty} \frac{1/n}{1} = 0$.
Find $\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$.
- This is the definition of $e$.
- $\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e$.
📌 Monotonic Sequences and Boundedness
| Type | Definition |
|---|
| Increasing | $a_{n+1} \ge a_n$ for all $n$ |
| Decreasing | $a_{n+1} \le a_n$ for all $n$ |
| Monotonic | Either increasing or decreasing |
| Bounded above | There exists $M$ such that $a_n \le M$ for all $n$ |
| Bounded below | There exists $m$ such that $a_n \ge m$ for all $n$ |
| Bounded | Both bounded above and below |
📌 Monotonic Sequence Theorem
If a sequence is both monotonic (increasing or decreasing) and bounded, then it converges.
Show that $a_n = \frac{1}{n}$ is monotonic and bounded, and therefore converges.
- Decreasing: $\frac{1}{n+1} < \frac{1}{n}$
- Bounded below by $0$, above by $1$
- Converges to $0$
⚠️ Common Mistakes
- Confusing sequences with series: A sequence is a list; a series is a sum.
- Assuming a sequence diverges if it oscillates: Not all oscillating sequences diverge (e.g., $(-1)^n/n$ converges to $0$).
- Forgetting to check boundedness: A monotonic sequence that is not bounded will diverge.
🔍 Key Takeaways
- A sequence $\{a_n\}$ converges if $\lim_{n \to \infty} a_n$ exists (finite).
- Use limit laws, L'Hôpital, and known limits to evaluate sequence limits.
- A monotonic bounded sequence always converges (Monotonic Sequence Theorem).
- Sequences are the building blocks for infinite series.
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