Calculus: Early Transcendentals (7th Edition): From Lists to Limits: The Foundation of Sequences

Imagine the universe as a series of snapshots. A sequence is exactly that: an ordered list of real numbers where the position (the index $n$) defines the value. Unlike a set, order and repetition are the heartbeat of the structure.

1. The Rigorous Definition

A sequence $\{a_n\}$ can be thought of as a list: $a_1, a_2, a_3, \dots, a_n, \dots$. More formally, it is a function whose domain is the set of positive integers.

Definition 1 (Informal)

A sequence has the limit $L$ (written $\lim_{n \to \infty} a_n = L$) if we can make the terms $a_n$ as close to $L$ as we like by taking $n$ sufficiently large.

Definition 2 (Formal ε-N)

$\lim_{n \to \infty} a_n = L$ if for every $\varepsilon > 0$ there is a corresponding integer $N$ such that if $n > N$ then $|a_n - L| < \varepsilon$.

2. The Bridge to Calculus: Theorem 3

One of our most powerful tools is the ability to treat discrete sequences as continuous functions. This allows us to use the full weight of L'Hospital's Rule.

If $\lim_{x \to \infty} f(x) = L$ and $f(n) = a_n$, then $\lim_{n \to \infty} a_n = L$.

Worked Example

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

Consider $f(x) = \frac{\ln x}{x}$. As $x \to \infty$, we have an $\infty/\infty$ indeterminate form. Applying L'Hospital's Rule:

$\lim_{x \to \infty} \frac{1/x}{1} = 0$. By Theorem 3, the sequence also converges to 0.

3. Divergence Nuance

Divergence isn't always about "blowing up" to infinity. A sequence can diverge through oscillation. Consider $a_n = (-1)^n$. The terms bounce forever between $-1$ and $1$, never settling on a single value.

🎯 Core Principle

Convergence requires that for any tiny distance ε you choose, there is a point in the sequence (N) after which all remaining terms are trapped within that distance from the limit L.

Thematic sidebar: In the last section of this chapter you are asked to use a series to derive a formula for the velocity of an ocean wave.

QUESTION 1

What is the general term formula $a_n$ for the sequence $\{3/5, 4/25, 5/125, 6/625, 7/3125, \dots\}$?

$a_n = (n+2)/5^n$

$a_n = (n+1)/5^n$

$a_n = (n+2)/5^{n-1}$

$a_n = 3n/5^n$

QUESTION 2

Which statement correctly describes the convergence of $a_n = 1 - (0.2)^n$?

It converges to 1 because $(0.2)^n \to 0$ as $n \to \infty$.

It diverges because $0.2 < 1$.

It converges to 0.

It diverges by oscillation.

QUESTION 3

What is the first term $a_1$ of the sequence $a_n = 2n/(n^2 + 1)$?

0.5

1.5

QUESTION 4

According to Theorem 3, why can we use L'Hospital's Rule on sequences?

Because if the continuous function $f(x)$ has a limit $L$, its integer samples must also approach $L$.

Because all sequences are differentiable.

Because sequences are the same as continuous functions.

Because L'Hospital's Rule only applies to discrete indices.

QUESTION 5

Which of these is a correct formula for $\{1, 1/3, 1/5, 1/7, 1/9, \dots\}$ starting at $n=1$?

$a_n = 1/(2n-1)$

$a_n = 1/(2n+1)$

$a_n = 1/n^2$

$a_n = 2/n$