MATH40002 Set4 REVISED
English review edition prepared on 4 October 2026 from a preserved source copy. It is a later presentation, not the historical study interface. Source checks and difficulty judgements describe the original material author's own process; they do not indicate Imperial College London endorsement.
Source SHA-256: cc28ffe6cc7ec48bf7ee60c9a3cf9d6b2a9c73d66333126b769c97726ce738c5Source date: 2026-08-06
Source page 1
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (19% down the source page)
The main solutions prioritise methods actually used in the official answers accessible during preparation.
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No lecture notes, problem sheets, tutorials or marked scripts from Tony were accessible. This booklet does not invent their contents, and advanced alternative methods are not the sole standard for solutions.
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Recognition signal
English passage 4 (42% down the source page)
For cardinality classify infinitely many free coordinates, eventually fixed coordinates, and fixed tails. Read quantifiers layer by layer; prove a supremum in two steps; choose an unbounded subsequence whose absolute values successively exceed k.
English passage 5 (47% down the source page)
First key step
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Write T = ∪U ; for sup prove an upper bound and then ε-approximation; recursively choose indices for the unbounded subsequence.
English passage 7 (75% down the source page)
Cardinality classification: 5 marks; interpreting quantifiers: 4; supremum proof: 4; unbounded subsequence: 7.
English passage 8 (78% down the source page)
Common errors
English passage 9 (81% down the source page)
Common mistakes: assume all infinite sequences over a finite alphabet form a countable set; misread whether a depends on N; select only values above k and ignore negative unboundedness.
English passage 10 (86% down the source page)
Final self-check
English passage 11 (89% down the source page)
For unboundedness use |a_n|. U_N has N − 1 free coordinates.
Source page 2
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Tony study template card
English passage 2 (12% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 3 (14% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 4 (30% down the source page)
Recognition signal
English passage 5 (33% down the source page)
Obtain a telescoping lower bound by rationalising a square-root difference. For exponential terms with a_n → a < 1, use eventual geometric comparison. Split a power series defined by parity into its even and odd subseries.
English passage 6 (38% down the source page)
First key step
English passage 7 (41% down the source page)
First rationalise; choose A ∈ (a, 1); separate the even and odd terms.
English passage 8 (76% down the source page)
Square-root inequality: 2 marks; divergence from first principles: 6; eventual geometric comparison: 6; ordinary series and radius: 6.
English passage 9 (79% down the source page)
Common errors
English passage 10 (81% down the source page)
Common mistakes: reverse an inequality; cite the p-series test despite the first-principles requirement; misread the exponent on a_n.
English passage 11 (81% down the source page)
Do not mistake a_n^n for n a_n, or apply the alternating test solely because an expression alternates.
English passage 12 (85% down the source page)
Final self-check
English passage 13 (88% down the source page)
Divergence at z = 1 suffices for R ≤ 1; inside the radius, absolute convergence must be proved.
Source page 3
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Tony study template card
English passage 2 (12% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 3 (14% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 4 (32% down the source page)
Recognition signal
English passage 5 (35% down the source page)
For oscillation multiplied by a power at 0, use squeezing and the difference quotient; for a sign change use IVT to obtain a root; the second derivative sign gives convex/concave midpoint inequalities; for multiple zeros repeatedly apply
English passage 6 (40% down the source page)
First key step
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First bound | sin | ≤ 1; construct a continuous difference function; calculate the sign of h; decrease the zero count level by level.
English passage 8 (75% down the source page)
Continuity/differentiability: 5 marks; IVT and zeros: 5; Taylor + concavity: 6; repeated Rolle: 4.
English passage 9 (78% down the source page)
Common errors
English passage 10 (81% down the source page)
Common mistakes: apply the chain rule directly at x = 0; allow endpoint zeros despite an open-interval requirement; reverse the convex/concave inequality; undercount zeros when repeatedly using Rolle.
English passage 11 (86% down the source page)
Final self-check
English passage 12 (89% down the source page)
h should be concave, so its midpoint value exceeds the average endpoint value.
Source page 4
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Tony study template card
English passage 2 (12% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 3 (14% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 4 (30% down the source page)
Recognition signal
English passage 5 (33% down the source page)
For Darboux sums on a special partition use monotonicity to choose endpoints; compare improper integrals with 1/xr; for power-series orthogonality substitute the truncated polynomial
English passage 6 (35% down the source page)
and then use uniform convergence to obtain the integral of the square.
English passage 7 (38% down the source page)
First key step
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write xi = ei/n; use ey ≥1 + y and MVT to squeeze; let PN is Taylor truncation .
English passage 9 (74% down the source page)
Upper and lower sums: 8 marks; improper integral: 6; power-series orthogonality: 6.
English passage 10 (77% down the source page)
Common errors
English passage 11 (80% down the source page)
Common mistakes: forget that the Darboux interval lengths are not a constant 1/n; state asymptotic inequalities for r > 1 without proof; interchange integral and limit without explaining uniform convergence.
English passage 12 (85% down the source page)
Final self-check
English passage 13 (88% down the source page)
U −L should simplify to (e −1)/n;
Source page 5
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
English passage 1 (9% down the source page)
Tony study template card
English passage 2 (12% down the source page)
Recognise the signal → write the first key step → check the theorem hypotheses → complete the course-method sequence → verify using the self-check.
English passage 3 (14% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.