MAST10006 · Calculus 2
Sequences and Series
A sequence (an) is an ordered list; a series ∑an is what you get by adding them all up. The central question for both is the same — does it converge? — and sequences come first because a series is defined as the limit of its sequence of partial sums. This chapter covers sequence limit laws and the standard sequence limits (including the sign trap in rn), the monotone-plus-bounded convergence guarantee that proves convergence without finding the limit, the geometric series (almost the only series you can sum in closed form) and the p-series cutoff, and then the full convergence-test toolkit. The signature exam task is “determine whether ∑an converges”: there is no single algorithm — you read the shape of the terms and pick the matching test, then check its conditions, because the marks are in the justified choice, not the one-word verdict.
What this chapter covers
- 012.1 Limit of a sequence — laws and standard limits
- 02Monotone + bounded ⇒ converges (without finding the limit)
- 033.1 The geometric series and the p-series cutoff
- 043.2 The divergence (nth-term) test — do this first
- 05The convergence-test toolkit and the decision tree
- 06Absolute vs conditional convergence
Worked example: choose and apply a convergence test
- +1Read the shape: a factorial over a power → the ratio test is the natural choice.
- +1Form the ratio: an+1/an = [(n+1)! / (n+1)n+1] · [nn / n!] = nn / (n+1)n = (n / (n+1))n.
- +1Take the limit: (n/(n+1))n = (1 + 1/n)−n → e−1 ≈ 0.37.
- +1Compare to 1: L = e−1 < 1, so by the ratio test the series converges (absolutely).
Key terms
- Convergent sequence
- A sequence (an) whose terms approach a single finite limit L as n → ∞. The sequence limit laws mirror those for functions, and the bridge theorem lets you apply L'Hopital to the continuous version f(x) and read the sequence limit off the result.
- Monotone convergence theorem
- An increasing sequence bounded above converges, and a decreasing sequence bounded below converges — even when you cannot find the limit directly. Especially useful for recursively defined sequences: prove monotone, prove bounded, conclude convergence, then take limits of the recurrence to get the value.
- Geometric series
- ∑ arn sums to a/(1 − r) exactly when |r| < 1, and diverges for |r| ≥ 1, where a is the first term actually present. Almost the only series you can sum in closed form, and a universal comparison yardstick.
- p-series
- ∑ 1/np, which converges if and only if p > 1. The harmonic series (p = 1) diverges even though 1/n → 0. Together with the geometric series, the p-series is the standard thing to compare against.
- Conditional convergence
- A series ∑an that converges, but whose absolute version ∑|an| diverges — the alternating harmonic series ∑(−1)n+1/n is the classic example. Contrast absolute convergence, where ∑|an| itself converges.
Sequences and Series FAQ
What is the hardest part of series questions?
Choosing the right test. There is no single algorithm — you read the shape of an and match it to a test: factorials or exponentials point to the ratio test, an nth power to the root test, a polynomial ratio to limit comparison with a p-series, an (−1)n to the alternating test. The marks are in the justified choice plus the condition check, not the one-word verdict.
If an → 0, does the series converge?
No — this is a one-directional test. If an does NOT tend to 0 the series diverges, but if an → 0 the divergence test says nothing: the series might converge or diverge. The harmonic series is the proof: 1/n → 0 yet ∑1/n diverges. Run the divergence test first as a filter, then move to a real test.
When are the ratio and root tests inconclusive?
At L = 1 both tell you nothing — you must switch to another test. This always happens on p-series, where the ratio tends to 1, so reach for comparison, the integral test or (for alternating series) the Leibniz test instead. Never conclude anything from L = 1.
How do I prove a recursive sequence converges without knowing its limit?
Use the monotone convergence theorem: show the sequence is monotone (examine an+1 − an or the ratio an+1/an) and bounded (find a wall it cannot pass). Those two facts force convergence. Then, if you want the value, set L = lim an = lim an+1, substitute into the recurrence, solve, and discard any root that violates the bounds.
Exam move
Internalise the decision tree, because the exam question is almost always “does ∑an converge?” First run the divergence test as a free filter. Then read the shape of the terms: exactly geometric or p-series → quote the rule; a polynomial ratio → limit comparison with the dominant power; a factorial or exponential → ratio test; an nth power → root test; an alternating sign → Leibniz, then test for absolute convergence. Whatever test you pick, state its conditions on the page — the integral test needs a positive, continuous, decreasing function; comparison needs non-negative terms with the inequality pointing the right way; alternating needs bn decreasing to 0. For sequences, lean on the bridge theorem to borrow L'Hopital, and use monotone-plus-bounded for recurrences. The verdict is worth little; the justified method is worth the marks.
