Everything from Week 2 onward has been a limit of some kind: of a difference quotient, of a Riemann sum, of an integral. This week's limits are of a sequence — a list of numbers indexed by the positive integers — and then of a running total of infinitely many terms.
14.1 Sequences
A sequence {an} is a function whose domain is the positive integers, written a1, a2, a3, … A sequence can be given by an explicit formula (an=1/n) or recursively (a1=1, an+1=an/2).
14.2 The Limit of a Sequence
{an} converges to L if an can be made arbitrarily close to L by taking n sufficiently large — the same idea as Week 3's limits at infinity (§3.8), restricted to integer inputs:
If no such L exists, {an} diverges. A powerful shortcut: if an=f(n) for a function f defined on the reals with limx→∞ f(x) = L, then limn→∞ an = L as well — every technique from Weeks 3 and 12 (L'Hôpital's Rule included) becomes available for sequences.
Example 14.1 — Sequence limit via L'Hôpital's Rule
Find limn→∞ (ln n)/n.
Step 1 — Replace n with the continuous variable x
Consider f(x) = (ln x)/x, an ∞/∞ form as x→∞.
Step 2 — Apply L'Hôpital's Rule (Week 12, §12.2)
limx→∞ (1/x)/1 = 0.
Step 3 — Transfer the result back to the sequence
limn→∞ (ln n)/n = 0
14.3 Sequence Limit Laws and the Squeeze Theorem
The limit laws from Week 2 §2.6 (sum, product, quotient, constant multiple) all hold for sequences unchanged, as does the Squeeze Theorem (Week 2 §2.9): if an ≤ bn ≤ cn and an, cn both converge to L, then bn converges to L too.
14.4 Bounded and Monotonic Sequences
{an} is bounded above if some number M satisfies an ≤ M for every n (bounded below, similarly); it is monotonic if it is entirely non-decreasing or entirely non-increasing — exactly Week 6 §6.6's language, applied to a sequence instead of a continuous function.
14.5 The Monotonic Sequence Theorem
If {an} is monotonic and bounded, it converges. This is the discrete counterpart of the Extreme Value Theorem's underlying idea (Week 6 §6.2): a sequence that keeps moving in one direction but never escapes a bounded region has nowhere left to go but toward a limit.
14.6 Infinite Series and Partial Sums
Given a sequence {an}, the infinite series ∑n=1∞ an is defined via its partial sums SN = ∑n=1N an (compare Week 7's sigma notation, §7.3):
The series converges to that limit if it exists (finite), and diverges otherwise — the same convergence language as Week 13's improper integrals, and not a coincidence: both add up infinitely many pieces by taking a limit of a finite truncation.
14.7 The Geometric Series
The geometric series ∑n=0∞ arn (first term a, common ratio r) is the one series whose convergence can be determined completely and whose sum can be found exactly:
The finite partial sum SN = a(1−rN+1)/(1−r) is provable by the standard "multiply by r and subtract" trick; letting N→∞ sends rN+1→0 exactly when |r|<1.
Example 14.2 — Summing a geometric series
Evaluate ∑n=0∞ 3(1/2)n.
Step 1 — Identify a and r, and confirm convergence
a=3, r=1/2, |r|<1, so the series converges (Eq. 14.3).
Step 2 — Apply the sum formula
3/(1−1/2) = 3/(1/2) = 6
14.8 The nth-Term Test for Divergence
If ∑ an converges, then necessarily limn→∞ an = 0 — the terms being added must eventually shrink toward nothing, or the running total could never settle. The contrapositive gives a quick divergence test:
The converse is false: an→0 does not guarantee convergence. Week 15 §15.1 exhibits the harmonic series ∑ 1/n, whose terms → 0 while the series itself diverges — the nth-term test only ever proves divergence, never convergence.
Example 14.3 — The nth-term test
Show that ∑n=1∞ n/(n+1) diverges.
Step 1 — Find the limit of the terms
limn→∞ n/(n+1) = 1 (Week 3, §3.9 degree-comparison shortcut).
Step 2 — Apply Eq. 14.4
Since the limit is 1 ≠ 0, the series diverges.
14.9 Worked Capstone Example
Example 14.4 — A repeating decimal as a geometric series
Express 0.45 = 0.454545… as a fraction, using the geometric series.
Step 1 — Write the repeating decimal as a series
0.454545… = 0.45 + 0.0045 + 0.000045 + … = ∑n=0∞ 0.45(0.01)n.
Step 2 — Identify a, r and confirm convergence
a=0.45, r=0.01, |r|<1.
Step 3 — Apply Eq. 14.3
0.45/(1−0.01) = 0.45/0.99 = 45/99
= 5/11 (reduced), so 0.454545… = 5/11 exactly.
14.10 Nomenclature
| Symbol | Meaning |
|---|---|
| {an} | A sequence, a function on the positive integers |
| SN | The Nth partial sum of a series |
| geometric series | ∑arⁿ, with first term a and common ratio r |
| nth-term test | If a_n does not tend to 0, ∑a_n diverges |
14.11 Problems
- Sequence limit. Find limn→∞ (3n²+1)/(n²−2), using Week 3's degree-comparison shortcut.
- Sequence limit, L'Hôpital's. Find limn→∞ n/en.
- Monotonic and bounded. Show that an=n/(n+1) is increasing and bounded above by 1, then state what the Monotonic Sequence Theorem concludes.
- Geometric series, converges. Evaluate ∑n=1∞ 4(−1/3)n.
- Geometric series, diverges. Explain why ∑n=0∞ 2(3/2)n diverges, using Eq. 14.3.
- nth-term test. Show that ∑n=1∞ cos(1/n) diverges.
- Repeating decimal. Express 0.7 as a fraction using the geometric series.
- Combined. A ball dropped from 10 m rebounds to 60% of its previous height each bounce. Using a geometric series, find the total distance it travels (up and down) before coming to rest.
14.12 References
- Thomas, G. B., Weir, M. D. and Hass, J. Thomas' Calculus, SI Units, 15th ed. Pearson, 2024. — §9.1 (sequences), §9.2 (infinite series).
- Pearson MyLab Math — the Week 14 homework set is assigned directly from the sections above; see the Calendar for the due date.