To determine if a sequence converges or diverges, you evaluate the limit of its terms as the index approaches infinity.
A sequence is an ordered list of numbers, often generated by a specific formula for its n-th term, denoted as $A_n$. For example, the sequence $A_n = 1/n$ produces the list $1, 1/2, 1/3, 1/4, \dots$. The core task is to ascertain whether these numbers approach a specific value or behave unpredictably as you progress further down the list.
The Limit Test: The Core Method
The most straightforward and widely used technique to ascertain a sequence's divergence or convergence is the Limit Test for Sequences.
Here's how to apply it:
- Identify the General Term ($A_n$): Pinpoint the algebraic expression or formula that defines the n-th term of your sequence.
- Compute the Limit: Calculate the limit of this general term ($An$) as the index n tends towards infinity:
$$ \lim{n \to \infty} A_n $$
Interpreting the Outcome:
- Convergence: If the computed limit equals a finite constant number (L), then the sequence is said to converge to L. This means that as n gets very large, the terms of the sequence get arbitrarily close to L. Examples of such constant numbers include $0, 1, \pi,$ or $-33$.
- Condition: $\lim_{n \to \infty} A_n = L$ (where L is a real number).
- Divergence: If the limit does not exist, or if it approaches positive infinity ($+\infty$) or negative infinity ($-\infty$), then the sequence diverges. This indicates that the sequence's terms do not settle on a specific value; instead, they might grow without bound, shrink without bound, or oscillate without approaching a single point.
- Condition: $\lim_{n \to \infty} A_n = \pm\infty$ or the limit does not exist.
Practical Examples for Clarity
Let's explore some common sequence types to illustrate convergence and divergence:
1. Example of a Convergent Sequence
Consider the sequence defined by $A_n = \frac{1}{n}$.
- Terms: The sequence starts with $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$
- Limit Calculation:
$$ \lim_{n \to \infty} \frac{1}{n} = 0 $$ - Conclusion: Since the limit is a finite number (0), this sequence converges to 0. As n becomes very large, the value of $1/n$ becomes increasingly close to zero.
2. Example of a Divergent Sequence (Approaching Infinity)
Consider the sequence defined by $A_n = 2n - 1$.
- Terms: The sequence generates $1, 3, 5, 7, \dots$
- Limit Calculation:
$$ \lim_{n \to \infty} (2n - 1) = \infty $$ - Conclusion: Because the limit is positive infinity, the sequence diverges. Its terms continue to grow without any upper bound.
3. Example of a Divergent Sequence (Oscillating)
Consider the sequence defined by $A_n = (-1)^n$.
- Terms: The sequence produces $-1, 1, -1, 1, \dots$
- Limit Calculation: As n approaches infinity, the terms of this sequence continuously alternate between -1 and 1. They never settle on a single value. Therefore, the limit does not exist.
- Conclusion: This sequence diverges due to its oscillating behavior.
Summary Table: Convergence vs. Divergence of Sequences
| Condition for $\lim_{n \to \infty} A_n$ | Sequence Behavior | Outcome |
|---|---|---|
| Equals a finite real number (L) | Terms get arbitrarily close to L | Converges to L |
| Equals $\pm\infty$ | Terms grow or shrink without bound | Diverges |
| Does not exist (e.g., oscillates) | Terms do not approach a single value | Diverges |
Key Considerations for Sequence Analysis
- Continuity Analogy: Often, if you can replace n with a continuous variable x and the function $f(x)$ has a limit as $x \to \infty$, then the sequence $A_n = f(n)$ will have the same limit. This allows using techniques from continuous functions.
- L'Hôpital's Rule: If the limit evaluation results in an indeterminate form (like $\frac{0}{0}$ or $\frac{\infty}{\infty}$), L'Hôpital's Rule can be applied after transforming the sequence expression into a continuous function.
- Squeeze Theorem: For more complex sequences, the Squeeze Theorem can be a powerful tool. If you can show that your sequence is "squeezed" between two other sequences that both converge to the same limit, then your sequence must also converge to that limit.
Understanding sequence convergence is a fundamental building block in calculus, laying the groundwork for the study of infinite series and their numerous applications in fields ranging from physics to finance. For further exploration, resources like Khan Academy offer comprehensive lessons and practice.
