MATH40002 Set1 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: f47a6eaa98dfceec3de580380e4ed762da23ca6ccc2a8dfdae29003ddddd5489Source 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
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The same positive sequence appears in sequence limits, ordinary series, alternating series and power series. First look for explicit differences, monotonicity and partial sums.
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First key step
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First write a_n = log(1 + 1/n). Immediately check whether the ordinary series telescopes; on the unit circle, check whether the geometric partial sums are bounded.
Source page 2
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
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Monotonicity and limit: 3 marks; use MVT to locate ξ_n and complete the ε–N proof: 3; telescoping partial sums: 3; alternating test and remainder: 5; radius and boundary analysis:
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Dirichlet 6 marks .
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Common errors
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Common mistakes: merely write a_n ∼ 1/n without the required ε–N proof; incorrectly claim absolute convergence on |z| = 1; fail to show that the geometric partial sums are bounded when z ≠ 1.
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Final self-check
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check PN
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n=1 an = log(N + 1); check that the boundary absolute value remains an.
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Tony study template card
English passage 9 (34% 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 10 (36% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 11 (52% down the source page)
Recognition signal
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Values specified on a dense set suggest choosing a sequence from that set. Finite limits at open endpoints suggest extending to the endpoints and using Heine–Cantor.
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Rapid oscillation suggests using two sequences to disprove uniform continuity.
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First key step
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Respectively: take a sequence from the dense set; define the endpoint values; construct two sequences whose distance tends to zero but whose function values differ by a fixed amount.
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 (19% down the source page)
Density and continuity: 5 marks; endpoint extension and proof at both ends: 4; Heine–Cantor: 4; continuous boundedness: 2; two-sequence counterexample: 5.
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Common errors
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Common mistakes: state density of the rationals/irrationals without choosing convergent sequences; assume continuity automatically gives uniform continuity; choose two sequences with equal function values, or whose distance does not tend to zero.
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Final self-check
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Check that both sequences lie in the domain, approach the same endpoint, and have function values differing by exactly 2.
English passage 6 (36% down the source page)
Tony study template card
English passage 7 (38% 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 8 (40% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 9 (56% down the source page)
Recognition signal
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For true/false questions, first check closure properties, convex/concave inequality directions, the chain rule, and whether twice differentiability requires a first derivative throughout a neighbourhood. State every theorem hypothesis.
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First key step
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First try simple affine, densely piecewise or oscillatory counterexamples. For an inverse-function question, start with h ◦ f = id.
Source page 4
Mathematical typesetting and diagrams are retained below. Chinese prose is replaced by the English passages that follow, in source reading order.
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True/false conclusions with reasons: 12 marks; three precise definitions or theorem statements: 8.
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Common errors
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Common mistakes: give True/False without proof; incorrectly extend preservation of convexity under maxima to concavity; infer twice differentiability merely from existence of u′(0); use the wrong order or coefficient in the Taylor remainder.
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Final self-check
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Check each theorem's domain, orders of continuity and differentiability, quantifier order, and integral-remainder coefficient.
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Tony study template card
English passage 7 (56% 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 8 (58% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
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Recognition signal
English passage 10 (76% down the source page)
Finitely many bad points suggest covering them by intervals of small total length. A fixed second-derivative sign and a bounded function suggest a monotone derivative.
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Rule out a nonzero slope by contradiction, then use an integral squeeze.
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First key step
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For integrability, first write |f| ≤ M and split into good and bad regions. For convex/concave asymptotics, first use f′′ to determine the monotonicity of f′.
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 (47% down the source page)
Bad-region length and uniform continuity on good regions: 4 marks; monotonicity: 3; a supremum-definition proof of the limit: 3; derivative limit: 4; integral squeeze: 4;
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final squeeze 2 marks .
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Common errors
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Common mistakes: omit interval length in the bad-region oscillation estimate; incorrectly infer that f′ increases from f′′ ≤ 0; use a supremum without its approximation property; finally infer the product limit from the derivative limit alone.
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f ′(x) →0 directly infer xf ′(x) →0
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Final self-check
English passage 7 (63% down the source page)
Check every inequality direction against concavity. The final squeeze uses the function difference tending to zero, not the product-limit theorem.
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Tony study template card
English passage 9 (69% 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 10 (71% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.