Math topic中文

Range of convergence

Range of values of xx for which a standard series converges.

Practice questions

  1. 1Section A: Pure Mathematics 1 (a) Show that 11+x2−1−x2=12x2(1+x2+1−x2).\dfrac{1}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}=\dfrac{1}{2x^{2}}\big(\sqrt{1+x^{2}}+\sqrt{1-x^{2}}\big).…
  2. 2Express 5+x2(2+x)(1−x)2\dfrac{5 + x^2}{(2 + x)(1 - x)^2} in the form of A(2+x)+B(1−x)2\dfrac{A}{(2 + x)} + \dfrac{B}{(1 - x)^2}, where AA and BB…
  3. 3It is given that f(x)=ln⁡(1+2x2)\mathrm{f}(x) = \ln(1 + 2x^2). A polynomial p(x)=ax2+bx+cp(x) = ax^2 + bx + c is used to approximate f(x)\mathrm{f}(x)…
  4. 4TMJC JC1 Promo 9758/2019/Q6 Given that y=ln⁡(2−e2x)y = \ln\left(2 - e^{2x}\right), where x<12ln⁡2x < \tfrac{1}{2}\ln 2, show that ey[d2ydx2+(dydx)2]=−4e2x.e^y\left[\frac{d^2 y}{dx^2} + \left(\frac{dy}{dx}\right)^2\right] = -4e^{2x}.…
  5. 5Section A: Pure Mathematics 1 (a) Show that 11+x2−1−x2=12x2(1+x2+1−x2).\displaystyle \frac{1}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}=\frac{1}{2x^{2}}\left(\sqrt{1+x^{2}}+\sqrt{1-x^{2}}\right).…
  6. 6Expand 1−x21+3x3\dfrac{1 - x^2}{\sqrt[3]{1 + 3x}} in ascending powers of xx up to and including the term in x3x^3. (ii) State…
  7. 7A sequence of numbers, xnx_{n}, satisfy the relation xn+1=13(2xn+1xn2),n∈N+∪{0},0<x0<1.x_{n+1}=\tfrac{1}{3}\left(2x_{n}+\dfrac{1}{x_{n}^{2}}\right),\quad n\in\mathbb{N}^{+}\cup\{0\},\quad 0<x_{0}<1.…
  8. 8Using standard series from the List of Formula (MF26), expand ln⁡(k+2x)n\ln(k + 2x)^n in ascending powers of xx, as far as the…
  9. 9TMJC JC1 Promo 9758/2019/Q6 Given that y=ln⁡(2−e2x)y = \ln(2 - \mathrm{e}^{2x}), where x<12ln⁡2x < \frac{1}{2}\ln 2, show that ey[d2ydx2+(dydx)2]=−4e2x.\mathrm{e}^y \left[ \frac{\mathrm{d}^2 y}{\mathrm{d}x^2} + \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2 \right] = -4\mathrm{e}^{2x}.…
  10. 10Using standard series, find the Maclaurin series expansion of 11−a2x2\dfrac{1}{\sqrt{1 - a^2 x^2}} up to and including the te…