MATH40002 Set6 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: e2fe5f0b38a416bbbd2123064cbf24292ef633bbb22d3c856cec3368e608558bSource 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.
English passage 2 (21% down the source page)
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.
English passage 3 (39% down the source page)
Recognition signal
English passage 4 (42% down the source page)
Use an injection to prove a function space uncountable. For the supremum of quotients, first bound and then approximate. Split cases for the limit of a maximum. For summable upward errors, add the tail to construct a monotone sequence.
English passage 5 (47% down the source page)
First key step
English passage 6 (50% down the source page)
Construct fg; choose sequences approaching sup/inf; assume a ≥ b without loss of generality; define tn as the tail of a p-series.
English passage 7 (85% down the source page)
Uncountability: 4 marks; supremum of quotients: 5; maximum limit: 5; error-corrected sequence: 6.
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)
Common errors
English passage 2 (12% down the source page)
Common mistakes: interpret periodicity as simultaneous translation of both variables; fail to check whether inf B may be zero; use continuity of a composite when a first-principles proof is required; choose the wrong sign for the corrected sequence.
English passage 3 (17% down the source page)
Final self-check
English passage 4 (20% down the source page)
c_n must be nonnegative and monotonically decreasing; t_n − t_(n+1) = 1/n².
English passage 5 (24% down the source page)
Tony study template card
English passage 6 (26% 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 7 (28% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 8 (45% down the source page)
Recognition signal
English passage 9 (47% down the source page)
For the radius of the coefficientwise product, use two interior radii and geometric comparison; terms of a convergent series are bounded; a positive-term average includes at least the first-term harmonic lower bound;
English passage 10 (49% down the source page)
squeeze the geometric mean of finitely many consecutive factors.
English passage 11 (53% down the source page)
First key step
English passage 12 (55% down the source page)
Choose x < R1, y < R2 and r < xy; Give |an| find a uniform bound; write bn ≥a1/n.
English passage 13 (87% down the source page)
Hadamard-product radius: 6 marks; weighted absolute comparison: 3; divergence of the series of averages: 4; geometric means and the alternating test: 7.
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)
Common errors
English passage 2 (12% down the source page)
Proving only that the general term tends to 0 rather than convergence of the power series; treating af(n) as though it were a subsequence; omitting the monotonicity condition of the alternating test.
English passage 3 (16% down the source page)
Final self-check
English passage 4 (18% down the source page)
The comparison ratio r/(xy) must be strictly less than 1.
English passage 5 (22% down the source page)
Tony study template card
English passage 6 (25% 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 7 (27% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 8 (45% down the source page)
Recognition signal
English passage 9 (47% down the source page)
Use sup for a monotone one-sided limit; a difference function for a fixed point; the chain rule for the derivative of an even function; if a function has a finite limit and its derivative has a limit,
English passage 10 (49% down the source page)
argue by contradiction; use nearby large points to disprove uniform continuity of a polynomial.
English passage 11 (53% down the source page)
First key step
English passage 12 (56% down the source page)
Respectively write L = sup, h = f − x, f ′(x) = −f ′(−x), assume the derivative limit is nonzero, and choose n and n + 1/n.
English passage 13 (80% down the source page)
One-sided limit: 5 marks; fixed point: 3; even function: 3; derivative limit: 4; failure of uniform continuity: 5.
English passage 14 (83% down the source page)
Common errors
English passage 15 (85% down the source page)
Common mistakes: assume existence of the one-sided limit; use Rolle for an even function to obtain a zero derivative somewhere rather than at 0; assume f′ tends to zero automatically by interchanging limits.
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)
Final self-check
English passage 2 (12% down the source page)
For failure of uniform continuity, both points should tend to infinity while their distance tends to 0.
English passage 3 (16% down the source page)
Tony study template card
English passage 4 (18% 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 5 (20% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.
English passage 6 (36% down the source page)
Recognition signal
English passage 7 (39% down the source page)
Darboux sums with finitely many bad points, nonnegative integrals, uniform Taylor remainders, subtracting a chord and using Rolle, and asymptotics of integral averages are frequent methods from five different years.
English passage 8 (44% down the source page)
First key step
English passage 9 (47% down the source page)
For integrability split into good and bad regions; for Taylor state a remainder bound; for a chord subtract the linear function; for integral asymptotics rewrite the limit as a two-sided inequality.
English passage 10 (88% down the source page)
Integrability with finite discontinuities: 5 marks; nonnegative integral: 3; uniform Taylor convergence: 4; chord and Rolle: 4; integral asymptotics: 4.
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)
Common errors
English passage 2 (12% down the source page)
An off-by-one subscript in the Taylor remainder factorial; forgetting that endpoints are also chord intersections; ignoring the constant contribution from [0, N] in integral asymptotics.
English passage 3 (16% down the source page)
Final self-check
English passage 4 (18% down the source page)
The uniform Taylor error does not depend on x; after division by x2 the constant term tends to 0.
English passage 5 (22% down the source page)
Tony study template card
English passage 6 (25% 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 7 (27% down the source page)
Even if you cannot finish, write the definition, key theorem and first step to earn method marks.