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Exercises

  1. Assume that lim⁡n→∞an=L\lim\limits_{n\to\infty} a_n = L，Prove that

    lim⁡n→∞a1+2a2+⋯+nan1+2+⋯+n=L.\begin{equation*} \lim_{n\to\infty} \frac{a_1 + 2 a_2 + \cdots + n a_n}{1+2+\cdots+n} = L. \end{equation*}

  2. Let {xn}\{x_n\} be a sequence in (0,1)(0,1) whose terms are all distinct.

    a. Prove that: if there exists a∈(0,1)a \in (0,1) such that for all n∈N+n \in \mathbb{N}^+,

    xn+2−xnxn+1−xn∈(a,1),\begin{equation*} \frac{x_{n+2} - x_n}{x_{n+1} - x_n} \in (a,1), \end{equation*}

    then the sequence {xn}\{x_n\} is convergent.

    b. Let c∈(0,34)c \in \left(0,\frac{3}{4}\right), and let {xn}\{x_n\} satisfy x1∈(0,1)x_1 \in (0,1) and for all n∈N+n \in \mathbb{N}^+,

    xn+1=1−cxn2.\begin{equation*} x_{n+1} = 1 - c x_n^2. \end{equation*}

    Prove that {xn}\{x_n\} is convergent and find its limit.

  3. Assume that

    lim⁡n→+∞an=A,A∈R or A=+∞ or A=−∞.\begin{equation*} \lim_{n\to +\infty} a_n = A,\qquad A \in \mathbb{R}\ \text{or}\ A = +\infty\ \text{or}\ A = -\infty. \end{equation*}

    For each positive integer nn, let tn1,…,tnnt_{n1},\ldots,t_{nn} be nn non-negative real numbers satisfying

    tn1+⋯+tnn=1,\begin{equation*} t_{n1} + \cdots + t_{nn} = 1, \end{equation*}

    and for any fixed positive integer kk,

    lim⁡n→+∞tnk=0.\begin{equation*} \lim_{n\to+\infty} t_{nk} = 0. \end{equation*}

    Prove that

    lim⁡n→+∞(tn1a1+⋯+tnnan)=A.\begin{equation*} \lim_{n\to+\infty} \left(t_{n1}a_1 + \cdots + t_{nn}a_n\right) = A. \end{equation*}

  4. Assume that q(x)<0q(x) < 0 for all x∈(a,b)x \in (a,b). Consider the boundary value problem

    {y′′+p(x)y′+q(x)y=r(x),a<x<b,y(a)=A,y(b)=B.\begin{equation*} \begin{cases} y'' + p(x)y' + q(x)y = r(x), & a < x < b,\\[6pt] y(a) = A,\quad y(b) = B. \end{cases} \end{equation*}

    Prove that: If the boundary value problem has a solution, then this solution must be unique.

  5. Let ff be differentiable on [0,+∞)[0,+\infty), satisfying

    f(0)=0,f′ is strictly increasing.\begin{equation*} f(0)=0, \qquad f' \text{ is strictly increasing}. \end{equation*}

    Prove that the function

    f(x)x\begin{equation*} \frac{f(x)}{x} \end{equation*}

    is strictly increasing on (0,+∞)(0, +\infty).

  6. Let 0<a<b0 < a < b. Compare the sizes of the following three quantities:

    ab,b−aln⁡b−ln⁡a,a+b2.\begin{equation*} \sqrt{ab}, \qquad \frac{b-a}{\ln b - \ln a}, \qquad \frac{a+b}{2}. \end{equation*}

  7. Let ff be a C∞C^\infty function and define

    g(x)=f(ln⁡x).\begin{equation*} g(x) = f(\ln x). \end{equation*}

    Prove that for every positive integer nn,

    xndnydxn=(ddt−(n−1))⋯(ddt−1)ddtf(t),\begin{equation*} x^n \frac{d^n y}{dx^n} = \left( \frac{d}{dt} - (n-1) \right) \cdots \left( \frac{d}{dt} - 1 \right) \frac{d}{dt} f(t), \end{equation*}

    where

    t=ln⁡x.\begin{equation*} t = \ln x. \end{equation*}

  8. Define

    Pn(x)=exdndxn(e−xxn).\begin{equation*} P_n(x) = e^x \frac{d^n}{dx^n}\left( e^{-x} x^n \right). \end{equation*}

    Prove that:

Pn(x) has n distinct positive zeros.\begin{equation*} P_n(x) \text{ has } n \text{ distinct positive zeros.} \end{equation*}

  1. Compute the limit

    lim⁡x→01−cos⁡xcos⁡2x cos⁡3x3x2.\begin{equation*} \lim_{x \to 0} \frac{ 1 - \cos x \sqrt{\cos 2x}\, \sqrt[3]{\cos 3x} }{ x^2 }. \end{equation*}

  2. Evaluate the integral:

∫dxsin⁡3x+cos⁡3x.\begin{equation*} \int \frac{dx}{\sin^3 x + \cos^3 x}. \end{equation*}

  1. Evaluate the integral:

    ∫tan⁡(x)  2+4+cos⁡x  dx\begin{equation*} \int \tan(x)\,\sqrt{\,2 + \sqrt{4 + \cos x}\,}\,dx \end{equation*}

  2. Evaluate the integral:

    ∫0∞dx(x+1+⌊2x⌋)2\begin{equation*} \int_{0}^{\infty} \frac{dx}{\bigl(x + 1 + \lfloor 2\sqrt{x} \rfloor\bigr)^2} \end{equation*}

  3. Prove the following inequalities:

    a. For any x∈[0,π]x \in [0,\pi] and t∈[0,1]t \in [0,1], we have sin⁡(tx)≥tsin⁡x.\sin(tx) \ge t \sin x.

    b. For any x≥0x \ge 0 and p>0p>0, we have

    ∫0x∣sin⁡u∣p du≥x ∣sin⁡x∣pp+1.\begin{equation*} \int_{0}^{x} |\sin u|^{p}\, du \ge \frac{x\, |\sin x|^{p}}{p+1}. \end{equation*}

  4. Find an integrable function ff on [0,1][0,1] such that for every x∈[0,1]x \in [0,1].

    f(x)=1+(1−x)∫0xyf(y) dy+x∫x1(1−y)f(y) dy.\begin{equation*} f(x) = 1 + (1-x) \int_{0}^{x} y f(y)\, dy + x \int_{x}^{1} (1-y) f(y)\, dy . \end{equation*}
  5. ∫02πdθAcos⁡2θ+2Bcos⁡θsin⁡θ+Csin⁡2θ,A>0,  AC>B2\begin{equation*} \int_{0}^{2\pi} \frac{d\theta}{ A \cos^2 \theta + 2B \cos \theta \sin \theta + C \sin^2 \theta }, \qquad A > 0,\; AC > B^2 \end{equation*}
  6. ∫0a(a23−x13)32 dx\begin{equation*} \int_{0}^{a} \left( a^{\frac{2}{3}} - x^{\frac{1}{3}} \right)^{\frac{3}{2}} \, dx \end{equation*}
  7. lim⁡n→∞1ln⁡n∫0π/2sin⁡2(nx)sin⁡x dx.\begin{equation*} \lim_{n \to \infty} \frac{1}{\ln n} \int_{0}^{\pi/2} \frac{\sin^2 (n x)}{\sin x}\, dx. \end{equation*}