example-queries.js

exports.example_queries = [
  {
    'showcase':
      {
        'tags': ['proof', 'infinite series'],
        'content': 'Show that [imath]\\sum_{k=1}^\\infty \\dfrac{\\sin k^2} k[/imath] converges.'
      },
    'keywords': 'prove, $\\sum_{k=1}^\\infty \\dfrac{\\sin k^2} k$, convergence'
  },
  {
    'showcase':
      {
        'tags': ['convergence-divergence'],
        'content': 'Find whether [imath]\\sum\\limits_{n=1}^{\\infty} a^{1+\\frac1{2}+\\frac1{3}+\\dots+\\frac1{n}}[/imath] converges or not.'
      },
    'keywords': '$\\sum\\limits_{n=1}^{\\infty} a^{1+\\frac1{2}+\\frac1{3}+\\dots+\\frac1{n}}$, converges or not?'
  },
  {
    'showcase':
      {
        'tags': ['limits', 'calculus'],
        'content': 'What is this limit: [imath]\\lim_{n \\to \\infty} (1 + \\dfrac 1 n)^n[/imath] ?'
      },
    'keywords': 'limit, $\\lim_{n \\to \\infty} (1 + \\dfrac 1 n)^n$'
  },
  {
    'showcase':
      {
        'tags': ['general-relativity'],
        'content': 'What is this equation: [imath]H^2=\\frac{8\\pi G}{3}\\rho+\\frac{kc^2}{a^2}[/imath]'
      },
    'keywords': 'equation, $H^2=\\frac{8\\pi G}{3}\\rho+\\frac{kc^2}{a^2}$'
  },
  {
    'showcase':
      {
        'tags': ['polyhedra', 'analytic-geometry'],
        'content': 'Derive Euler\'s Formula for polyhedron: [imath]F-E+V=2[/imath]'
      },
    'keywords': '$F-E+V=2$, polyhedron'
  },
  {
    'showcase':
      {
        'tags': ['inequality'],
        'content': 'Find a tight constant lowerbound for [imath](xy+yz+xz)\\left(\\frac{1}{(x+y)^2}+\\frac{1}{(y+z)^2}+\\frac{1}{(x+z)^2}\\right)[/imath]'
      },
    'keywords': '$(xy+yz+xz)\\left(\\frac{1}{(x+y)^2}+\\frac{1}{(y+z)^2}+\\frac{1}{(x+z)^2}\\right)$'
  },
  {
    'showcase':
      {
        'tags': ['calculus', 'asymptotics'],
        'content': 'Prove Stirling\'s approximation [imath]n!\\sim \\sqrt {2\\pi n}(\\frac ne)^n[/imath]'
      },
    'keywords': 'proof, $n!\\sim \\sqrt {2\\pi n}(\\frac ne)^n$'
  }
]