What does it mean for a sequence to converge?

A sequence {aₙ} converges to L if, as n increases, aₙ can be made arbitrarily close to L. Formally, for every positive number ε, there exists an N such that |aₙ − L| < ε for all n ≥ N.

What is the difference between divergence and oscillation?

Divergence means the sequence grows without bound (e.g., aₙ = n). Oscillation means the sequence keeps bouncing between values without settling on any single limit (e.g., aₙ = (−1)ⁿ).

Chapter 3: Limits of Sequences

Q: What is the difference between divergence and oscillation?

Divergence means the sequence grows without bound (e.g., aₙ = n). Oscillation means the sequence keeps bouncing between values without settling on any single limit (e.g., aₙ = (−1)ⁿ).

What "approaching" really means

Goals of This Page

Develop an intuitive understanding of convergence, divergence, and oscillation
Prepare for the rigorous definition (the ε-N definition)

1. Intuitive Picture of Convergence

A sequence $\{a_n\}$ converges to $L$ if, as $n$ grows larger, $a_n$ can be made arbitrarily close to $L$.

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

$a_1 = 2$, $a_2 = 1.5$, $a_3 \approx 1.33$, $a_{10} = 1.1$, $a_{100} = 1.01$, ...

As $n$ increases, $a_n$ gets closer and closer to $1$. $\displaystyle\lim_{n \to \infty} \left(1 + \frac{1}{n}\right) = 1$

Figure 1: Convergence of $a_n = 1 + 1/n$ (all points lie within the ε-band for $n \geq N$)

2. What "Arbitrarily Close" Means

What does it mean concretely for $1 + \frac{1}{n}$ to approach $1$?

Want $|a_n - 1| < 0.1$? Achieved for $n \geq 11$.
Want $|a_n - 1| < 0.01$? Achieved for $n \geq 101$.
Want $|a_n - 1| < 0.001$? Achieved for $n \geq 1001$.

No matter how small a positive number $\varepsilon$ is specified, there always exists an $N$ such that $|a_n - 1| < \varepsilon$ holds for all $n \geq N$. This structure — "for every $\varepsilon$ there exists an $N$" — is the heart of the ε-N definition.

3. Divergence and Oscillation

Divergence: $a_n = n$

$a_1 = 1$, $a_2 = 2$, $a_3 = 3$, ...

As $n$ increases, $a_n$ grows without bound. It does not approach any value $L$.

Oscillation: $a_n = (-1)^n$

$a_1 = -1$, $a_2 = 1$, $a_3 = -1$, $a_4 = 1$, ...

The terms alternate between $-1$ and $1$, never settling on any single value.

Figure 2: Divergence (left) and oscillation (right)

Summary

Convergence: the terms can be made arbitrarily close to some value
Divergence: the terms grow without bound (or decrease without bound)
Oscillation: the terms never settle on a single value
Note: Strictly speaking, "divergence" means "failure to converge," so oscillation is a special case of divergence. Here we distinguish them for intuitive clarity.
The rigorous definition takes the form "for every $\varepsilon > 0$, there exists..."