Question 1 · transformed sequence and boundary series
Throughout this question, distinguish convergence of the positive series, convergence of the alternating series, and convergence of the associated complex power series. Any use of a convergence test must include its hypotheses, and every boundary point requested in the question must be treated separately.
Present the solution in the order of the labelled parts. Every final claim must be supported by a theorem, calculation or explicit check.
(a) For \(n\ge1\), define \(a_n=\sqrt{n+7}-\sqrt{n+2}\). Prove \(a_n>0\), show that \(a_n\to0\), and prove directly that \(\sqrt n\,a_n\to5/2\). (5 marks)
(b) Determine the convergence of \(\sum_{n\ge1}a_n\) using a valid partial-sum or comparison argument. (3 marks)
(c) Determine the convergence of \(\sum_{n\ge1}(-1)^{n-1}a_n\), and give an explicit condition on N making its truncation error below \(10^{-4}\). (5 marks)
(d) For \(F(z)=\sum_{n\ge1}a_nz^n\), find the radius of convergence and investigate separately z=1, z=-1 and \(z=e^{i\theta}\), \(0<\theta<2\pi\). (7 marks)
**[Total: 20 marks]**
Worked solution and marking guidance
**(a)** a_n=\frac5{\sqrt{n+7}+\sqrt{n+2}}. This proves positivity and, by the
displayed denominator/integral estimate, \(a_n\to0\). Moreover
\[
\sqrt n\,a_n=\frac5{\sqrt{1+7/n}+\sqrt{1+2/n}}\to\frac52.
\]
The same bounds give an explicit \(N(\varepsilon)\), so this is an
\(\varepsilon\)-\(N\) proof rather than a decimal approximation.
**(b)** For the partial sums,
\[
S_N=\sum_{j=N+3}^{N+7}\sqrt j-\sum_{j=3}^{7}\sqrt j\longrightarrow+\infty.
\]
Thus the positive series **diverges**. Equivalently, limit
comparison with \(a_n\sim\frac5{2\sqrt n}\) gives the same conclusion.
**(c)** The positive terms decrease to zero. Hence the alternating
series converges by the alternating-series theorem. Its remainder satisfies
\[
|S-S_N|\le a_{N+1},\qquad a_{N+1}\le\frac5{2\sqrt{N+3}}<10^{-4}; N+3>(25000)^2\text{ is sufficient}.
\]
**(d)** Polynomial decay of the coefficients gives
\(\limsup |a_n|^{1/n}=1\), hence the radius is \(R=1\).
Because \(\sum a_n\) diverges, \(z=1\) diverges; \(z=-1\) and every \(e^{i\theta}\ne1\) converge conditionally by the alternating/Dirichlet tests. No point on \(|z|=1\) is absolutely convergent.
Question 2 · density, compact extension and oscillation
Present the solution in the order of the labelled parts. Every final claim must be supported by a theorem, calculation or explicit check.
(a) Let \(f:[0,3]\to\mathbb R\) be continuous and suppose \(f(q)=q^3-2q\) for every rational \(q\in[0,3]\). Prove that \(f(x)=x^3-2x\) for every real x in the interval. (5 marks)
(b) Let \(g:(0,3)\to\mathbb R\) be continuous and suppose both finite one-sided endpoint limits exist. Construct a closed-interval extension, prove endpoint continuity and deduce uniform continuity of g. (7 marks)
(c) For \(h(x)=\sin(1/x)\) on (0,1), prove continuity and boundedness but disprove uniform continuity using two explicit sequences. The distance of the inputs must tend to zero while the outputs remain separated by a fixed amount. (6 marks)
(d) State a sharp standard hypothesis on the domain that turns continuity into uniform continuity, and name the theorem. (2 marks)
**[Total: 20 marks]**
Worked solution and marking guidance
**(a)** Fix \(x\in[0,3]\). Choose rationals \(q_n\to x\).
Continuity gives
\[
f(x)=\lim f(q_n)=\lim\bigl(q_n^3-2q_n\bigr)
=x^3-2x.
\]
**(b)** Let \(L_0=\lim_{x\to0^+}g(x)\) and
\(L_3=\lim_{x\to3^-}g(x)\). Define the extension by
\(G(0)=L_0\), \(G(3)=L_3\), and \(G=g\) inside.
The definitions of the one-sided limits prove endpoint continuity.
Thus \(G\) is continuous on compact \([0,3]\), so Heine-Cantor
makes \(G\), and therefore \(g\), uniformly continuous.
**(c)** The function is continuous and bounded by 1. Put
\[
x_n=(\pi/2+2\pi n)^{-1/1},\qquad
y_n=(3\pi/2+2\pi n)^{-1/1}.
\]
Then \(x_n,y_n\to0\), \(|x_n-y_n|\to0\), but
\(h(x_n)=1\) and \(h(y_n)=-1\). This contradicts uniform continuity.
**(d)** A continuous function on a compact metric domain is uniformly
continuous (Heine-Cantor theorem).
Question 3 · theorem hypotheses and decisive counterexamples
Treat the statements in the first part independently. A bare true/false answer earns no credit: for a true statement give a complete proof, while for a false statement specify all functions, domains and parameter values in a counterexample and verify that they satisfy the stated hypotheses. In the theorem-statement parts, write every quantifier and regularity assumption needed for the version you use; naming a theorem without its hypotheses is not a complete answer.
Present the solution in the order of the labelled parts. Every final claim must be supported by a theorem, calculation or explicit check.
(a) For each statement, decide true or false and give a proof or a fully specified counterexample: (i) the product of two Darboux-integrable functions is Darboux integrable; (ii) the minimum of two convex functions is always convex; (iii) a differentiable bijection with differentiable inverse has non-zero derivative everywhere. (9 marks)
(b) State the definition of Darboux integrability using infima, suprema, lower sums, upper sums and the complete epsilon quantifiers. (4 marks)
(c) State Taylor's theorem of order n with integral remainder and all regularity assumptions. (4 marks)
(d) State one valid version of L'Hôpital's rule, including the indeterminate forms and every essential hypothesis. (3 marks)
**[Total: 20 marks]**
Worked solution and marking guidance
**(a)** The answers are **True, False, True**.
(i) True: products of bounded Darboux-integrable functions are Darboux integrable (use the oscillation estimate for fg). (ii) False: f(x)=x and g(x)=-x are convex but \(\min(f,g)=-|x|\) is not convex. (iii) True: differentiating \(f^{-1}(f(x))=x\) gives \((f^{-1})'(f(x))f'(x)=1\), so \(f'(x)\ne0\).
**(b)–(d)**
For a partition \(P:a=x_0<\cdots<x_m=b\), put
\[
m_i=\inf_{[x_{i-1},x_i]}f,\quad M_i=\sup_{[x_{i-1},x_i]}f,
\quad L(f,P)=\sum m_i\Delta x_i,\quad U(f,P)=\sum M_i\Delta x_i.
\]
A bounded \(f\) is Darboux integrable iff
\(\sup_P L(f,P)=\inf_P U(f,P)\), equivalently iff for every
\(\varepsilon>0\) there is a partition \(P\) with
\(U(f,P)-L(f,P)<\varepsilon\).
If \(f^{(n)}\) is absolutely continuous (continuous
\(f^{(n+1)}\) is sufficient), then
\[
f(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k+
\frac1{n!}\int_a^x(x-t)^n f^{(n+1)}(t)\,dt.
\]
One standard L'Hopital form assumes differentiability on a punctured
interval, \(g'\ne0\), a \(0/0\) or \(\infty/\infty\) form, and an
existing limit of \(f'/g'\); under the standard endpoint hypotheses
the quotient has the same limit.
Question 4 · convex functions and decay at infinity
Monotonicity and limit conclusions must be derived from the sign of the second derivative and boundedness; do not assume in advance that the first derivative has a limit.
Present the solution in the order of the labelled parts. Every final claim must be supported by a theorem, calculation or explicit check.
(a) Let \(f:[0,\infty)\to\mathbb R\) be continuous, twice differentiable on (0,∞), bounded, and satisfy \(f''(x)\ge0\). Prove that f is non-increasing. (3 marks)
(b) With \(\ell=\inf_{x\ge0}f(x)\), prove directly from the definition of the extremum that \(f(x)\to\ell\) as x→∞. (4 marks)
(c) Prove \(f'(x)\to0\), explicitly ruling out every non-zero candidate limit. (4 marks)
(d) Prove \(2x f'(x)\le f(3x)-f(x)\le 2x f'(3x)\) and use it to show \(x f'(x)\to0\). (6 marks)
(e) Give a convex example showing which conclusion can fail when boundedness is removed. (3 marks)
**[Total: 20 marks]**
Worked solution and marking guidance
**(a)** Since \(f''\) has the stated sign, \(f'\) is
non-decreasing. The supporting
line inequality
\[
f(y)\ge f(x)+f'(x)(y-x)
\]
and boundedness rule out a derivative with the wrong strict sign.
Therefore \(f'\le0\) and \(f\) is non-increasing.
**(b)** With \(\ell=\inf f\), choose a point whose function value is
within \(\varepsilon\) of \(\ell\). Monotonicity traps every later
value between that value and \(\ell\), so \(f(x)\to\ell\).
**(c)** Monotonicity of \(f'\) gives a one-sided limit. The sign from
(a) and boundedness rule out every non-zero limit: integrating a
strictly positive or negative tail bound would make \(f\) unbounded.
Hence \(f'(x)\to0\).
**(d)** Monotonicity of \(f'\) on \([x,3x]\) and integration
give
\[
2x f'(x)\le f(3x)-f(x)\le 2x f'(3x).
\]
The middle increment tends to zero by (b). Applying the same estimate
on a rescaled preceding interval traps \(xf'(x)\) between 0 and a
vanishing function increment, so \(xf'(x)\to0\).
**(e)** Without boundedness, \(f(x)=x\) is convex but has a
non-zero constant derivative; the monotonicity conclusion in (a) and
the derivative/asymptotic conclusions can fail.