A sequence is an ordered list of numbers, often written as a1, a2, a3, and so on. In calculus, sequences matter because they show how quantities behave as the index n grows without bound. Some sequences settle closer and closer to a single number, while others grow, oscillate, or fail to approach any one value.
This idea is the foundation for limits, series, and many models of long-term behavior.
A sequence converges when its terms get arbitrarily close to a fixed number called the limit. It diverges if it does not approach a finite limit, such as when the terms grow without bound or keep jumping between values. Monotonic and bounded behavior gives an important test: a sequence that is always increasing and stays below some ceiling must converge.
Graphing sequence terms as discrete points helps students see the difference between approaching a horizontal limit line and escaping upward or oscillating.
Key Facts
- A sequence is a function whose input is a positive integer: a_n = f(n).
- A sequence converges to L if lim n→∞ a_n = L.
- A sequence diverges if lim n→∞ a_n does not exist as a finite number.
- Example of convergence: a_n = 1/n has lim n→∞ 1/n = 0.
- Example of divergence to infinity: a_n = n^2 has lim n→∞ n^2 = ∞.
- Monotone Convergence Theorem: if a sequence is monotonic and bounded, then it converges.
Vocabulary
- Sequence
- A sequence is an ordered list of numbers indexed by positive integers, such as a1, a2, a3, and so on.
- Limit of a sequence
- The limit of a sequence is the number the terms approach as n becomes very large.
- Convergent sequence
- A convergent sequence is a sequence whose terms approach a finite limit.
- Divergent sequence
- A divergent sequence is a sequence that does not approach a finite limit.
- Monotonic sequence
- A monotonic sequence is a sequence that is always nondecreasing or always nonincreasing.
Common Mistakes to Avoid
- Treating sequence graphs as continuous curves is wrong because sequences are defined only at integer values of n. Plot discrete points rather than connecting every value as if all real inputs are allowed.
- Assuming small terms always mean convergence is wrong because the terms must approach one fixed number. For example, a_n = (-1)^n stays bounded but does not converge.
- Confusing bounded with convergent is wrong because bounded sequences can still oscillate. A sequence must have terms that settle toward one value to converge.
- Using early terms to decide long-term behavior is wrong because convergence depends on what happens as n approaches infinity. Always analyze the formula or pattern for large n.
Practice Questions
- 1 Find the first five terms of a_n = 3 + 2/n, then determine lim n→∞ a_n.
- 2 Decide whether a_n = (5n + 1)/(2n - 3) converges, and if it does, find its limit.
- 3 A sequence is increasing and every term is less than 10. Explain why this information is enough to conclude that the sequence converges.