01 — What Is a Sequence?
A sequence is a function whose domain is the natural numbers: an ordered list of terms a₁, a₂, a₃, … each produced by a rule. We write it as {aₙ} or (aₙ) for short.
Three ways to specify a sequence:
- Explicit formula: aₙ = 1/n → 1, 1/2, 1/3, 1/4, …
- Recursive formula: a₁ = 1, aₙ = aₙ₋₁ + 2 → 1, 3, 5, 7, 9, …
- Listing the pattern: 1, 4, 9, 16, 25, … (recognise aₙ = n²)
02 — Convergence and Divergence
The most important question about any sequence: does it approach a finite limit as n grows without bound?
Formal Definition of Convergence
limn→∞ aₙ = L
means: for every ε > 0, there exists N such that for all n > N, |aₙ − L| < ε.
In plain English: we can get aₙ as close to L as we like, just by going far enough along the sequence.
Converges — the sequence has a finite limit L: lim aₙ = L.
Diverges to ±∞ — the terms grow (or shrink) without bound.
Diverges by oscillation — the terms bounce between values with no settled limit (e.g. (−1)ⁿ).
Diverges to ±∞ — the terms grow (or shrink) without bound.
Diverges by oscillation — the terms bounce between values with no settled limit (e.g. (−1)ⁿ).
03 — Limit Laws for Sequences
If lim aₙ = A and lim bₙ = B (both finite), the following hold:
Sequence Limit Laws
lim (aₙ ± bₙ) = A ± B
lim (aₙ · bₙ) = A · B
lim (aₙ / bₙ) = A / B [provided B ≠ 0]
lim (c · aₙ) = c · A [c any constant]
lim aₙᵖ = Aᵖ [provided Aᵖ is defined]
These mirror the limit laws for functions — if you know those, you know these.
A particularly powerful bridge: if f is continuous at A and aₙ → A, then f(aₙ) → f(A). This lets you "pass a limit inside" continuous functions like sin, cos, eˣ, and ln.
04 — Sequences vs. Functions: The Bridge Theorem
If a function f satisfies limx→∞ f(x) = L, then the sequence aₙ = f(n) also converges to L. This lets us use L'Hôpital's Rule and every limit technique from calculus directly on sequences.
Continuous Function Theorem for Sequences
If limx→∞ f(x) = L, then limn→∞ f(n) = L
The converse is false — a function may oscillate between integer values while still giving a converging sequence. Use this bridge in one direction only: function → sequence.
05 — Interactive Sequence Visualiser
Explore sequences: choose a type and watch convergence or divergence
CONVERGES
06 — Monotone Sequences and the Bounded Monotone Theorem
A sequence {aₙ} is monotone increasing if aₙ₊₁ ≥ aₙ for all n, and monotone decreasing if aₙ₊₁ ≤ aₙ. A sequence is bounded above if aₙ ≤ M for all n (and bounded below if aₙ ≥ m).
Monotone Convergence Theorem (MCT)
Every bounded monotone sequence converges.
• Bounded above + monotone increasing → converges to its supremum (least upper bound).
• Bounded below + monotone decreasing → converges to its infimum (greatest lower bound).
The MCT guarantees convergence without needing to find L explicitly — powerful when the limit is hard to compute directly.
Checking Monotonicity
Three methods to show {aₙ} is increasing:
- Direct: show aₙ₊₁ − aₙ ≥ 0
- Ratio: show aₙ₊₁/aₙ ≥ 1 (for positive sequences)
- Calculus: show f′(x) ≥ 0 where aₙ = f(n)
07 — The Squeeze Theorem for Sequences
Squeeze Theorem (Sequences)
If bₙ ≤ aₙ ≤ cₙ for all n ≥ N,
and lim bₙ = lim cₙ = L,
then lim aₙ = L.
and lim bₙ = lim cₙ = L,
then lim aₙ = L.
Classic use: bounding a complicated aₙ between two simpler sequences that share the same limit.
The Squeeze Theorem is especially useful for sequences involving oscillating factors like (−1)ⁿ, sin(n), or cos(n) that prevent direct computation of the limit.
08 — Seven Standard Limits to Know
| Sequence | Limit | Notes |
|---|---|---|
| 1/n | 0 | The fundamental null sequence |
| c^(1/n) (c > 0) | 1 | Any positive constant raised to the power 1/n → 1 |
| n^(1/n) | 1 | Even n itself, to the power 1/n, goes to 1 |
| xⁿ (|x| < 1) | 0 | Geometric decay to zero for |x| < 1 |
| xⁿ/n! | 0 | Factorial grows faster than any exponential |
| ln(n)/n | 0 | Logarithm is beaten by linear growth |
| (1 + 1/n)ⁿ | e | The definition of Euler's number e ≈ 2.71828 |
09 — Twelve Worked Examples
EX 01Direct limit — rational sequence
Find limn→∞ (3n² + 2n) / (5n² − 1).
1
Divide numerator and denominator by n² (highest power).
2
(3 + 2/n) / (5 − 1/n²)3
As n → ∞: 2/n → 0 and 1/n² → 0.
→ (3 + 0)/(5 − 0)lim = 3/5 — the sequence converges to 3/5.
EX 02Oscillating sequence — divergence
Does aₙ = (−1)ⁿ converge?
1
Odd terms: a₁ = −1, a₃ = −1, a₅ = −1, … → subsequence limit = −1.
2
Even terms: a₂ = 1, a₄ = 1, a₆ = 1, … → subsequence limit = +1.
3
Two subsequences have different limits, so the sequence cannot converge.
Diverges — oscillates between −1 and +1 forever.
EX 03Alternating convergence — aₙ = (−1)ⁿ/n
Find the limit of aₙ = (−1)ⁿ/n.
1
Bound it:
−1/n ≤ (−1)ⁿ/n ≤ 1/n2
Both bounds → 0 as n → ∞.
3
By the Squeeze Theorem: lim (−1)ⁿ/n = 0.
lim = 0 — the oscillations shrink fast enough to converge.
EX 04Exponential squeeze — aₙ = sin(n)/n
Find limn→∞ sin(n)/n.
1
Use the bound |sin(n)| ≤ 1:
−1/n ≤ sin(n)/n ≤ 1/n2
Both bounds → 0.
lim = 0 — squeeze theorem handles the wild oscillation of sin(n).
EX 05L'Hôpital via the bridge theorem
Find limn→∞ ln(n)/n.
1
Consider f(x) = ln(x)/x. As x → ∞ this is ∞/∞ form.
2
Apply L'Hôpital:
f(x) → (1/x)/1 = 1/x → 03
By the Bridge Theorem: lim ln(n)/n = 0.
lim = 0 — logarithm grows slower than any power of n.
EX 06Geometric sequence
Does aₙ = 3 · (0.8)ⁿ converge?
1
|r| = 0.8 < 1, so rⁿ → 0 (standard geometric limit).
2
lim 3 · (0.8)ⁿ = 3 · 0 = 0Converges to 0.
EX 07The definition of e
Show that aₙ = (1 + 1/n)ⁿ converges and state its limit.
1
Take ln: ln aₙ = n · ln(1 + 1/n). As n → ∞ this is ∞ · 0 form.
2
Rewrite:
ln(1 + 1/n) / (1/n), which is 0/0 form → apply L'Hôpital → limit of ln aₙ = 1.3
Since ln aₙ → 1, we have aₙ → e¹ = e (exponential is continuous).
lim (1 + 1/n)ⁿ = e ≈ 2.71828 — this is the definition of Euler's number.
EX 08Recursive sequence — MCT
a₁ = 1, aₙ₊₁ = ½(aₙ + 2). Show the sequence converges and find L.
1
Bounded above by 2: if aₙ ≤ 2 then aₙ₊₁ = ½(aₙ+2) ≤ ½(2+2) = 2. By induction aₙ ≤ 2 for all n ✓
2
Increasing: aₙ₊₁ − aₙ = ½(aₙ+2) − aₙ = (2−aₙ)/2 ≥ 0 since aₙ ≤ 2 ✓
3
By MCT, the limit L exists. Take limits of both sides:
L = ½(L + 2) → 2L = L + 2 → L = 2Converges to L = 2.
EX 09Factorial vs exponential
Does aₙ = 3ⁿ/n! converge?
1
Check the ratio:
aₙ₊₁/aₙ = [3ⁿ⁺¹/(n+1)!] / [3ⁿ/n!] = 3/(n+1)2
As n → ∞:
3/(n+1) → 0. So the ratio → 0 < 1.3
Eventually aₙ₊₁ < aₙ (decreasing, bounded below by 0) → converges.
Converges to 0 — factorial ultimately dominates any fixed exponential.
EX 10Telescoping limit
Find limn→∞ (√n − √(n−1)).
1
Rationalise: multiply by
(√n + √(n−1))/(√n + √(n−1))2
= (n − (n−1)) / (√n + √(n−1)) = 1/(√n + √(n−1))3
As n → ∞: denominator → ∞, so the fraction → 0.
lim = 0.
EX 11Divergence to +∞
Show aₙ = n² − 5n diverges to +∞.
1
Factor:
aₙ = n(n − 5). For n > 5 both factors are positive and growing.2
Lower bound: for n ≥ 10, n − 5 ≥ n/2, so aₙ ≥ n²/2 → ∞.
Diverges to +∞.
EX 12Subsequence argument
Show aₙ = cos(nπ) diverges.
1
cos(nπ) = (−1)ⁿ: +1 when n is even, −1 when n is odd.
2
The even subsequence → +1 and the odd subsequence → −1.
3
A convergent sequence must have all subsequences converging to the same limit.
Diverges — the two subsequences disagree.
10 — Common Mistakes
Mistake 1 — Confusing sequence convergence with boundedness: A bounded sequence need not converge (e.g. (−1)ⁿ is bounded but diverges). Boundedness is necessary for convergence, not sufficient.
Mistake 2 — Using L'Hôpital directly on sequences: L'Hôpital applies to continuous-variable functions. You must invoke the Bridge Theorem first (convert to f(x), apply L'Hôpital, conclude for the sequence).
Mistake 3 — Forgetting to check both sides of MCT: The Monotone Convergence Theorem requires both monotonicity and boundedness. A monotone unbounded sequence diverges (e.g. aₙ = n is increasing but unbounded → diverges).
Next step: Now that you understand individual terms, head to Series Convergence to learn what happens when you add them all up.
🔗 Related Topics — Series & Sequences
11 — Quiz: Sequences (10 Questions)
Question 1 of 10
Does aₙ = 1/n converge? If so, to what?
Question 2 of 10
limn→∞ (5n − 3) / (2n + 7) = ?
Question 3 of 10
Does aₙ = (−1)ⁿ converge?
Question 4 of 10
What does the Monotone Convergence Theorem require?
Question 5 of 10
limn→∞ (1 + 1/n)ⁿ = ?
Question 6 of 10
The Squeeze Theorem can be applied when…
Question 7 of 10
Does aₙ = 2ⁿ/n! converge?
Question 8 of 10
limn→∞ sin(n)/n = ?
Question 9 of 10
A sequence converges to L. What must be true of every subsequence?
Question 10 of 10
Which sequence diverges?
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