{
  "index": "1977-A-4",
  "type": "ANA",
  "tag": [
    "ANA",
    "ALG"
  ],
  "difficulty": "",
  "question": "Problem A-4\nFor \\( 0<x<1 \\), express\n\\[\n\\sum_{n=0}^{\\infty} \\frac{x^{2^{n}}}{1-x^{2^{n+1}}}\n\\]\nas a rational function of \\( x \\).",
  "solution": "A-4.\n\\[\n\\begin{aligned}\n\\sum_{n=0}^{N} \\frac{x^{2^{n}}}{1-x^{2^{n+1}}} & =\\sum_{n=0}^{N}\\left(\\frac{1}{1-x^{2^{n}}}-\\frac{1}{1-x^{2^{n+1}}}\\right) \\\\\n& =\\frac{1}{1-x}-\\frac{1}{1-x^{2^{N+1}}} \\rightarrow \\frac{1}{1-x}-1=\\frac{x}{1-x} \\text { as } N \\rightarrow \\infty\n\\end{aligned}\n\\]\nsince \\( |x|<1 \\).",
  "vars": [
    "x",
    "n",
    "N"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "variable",
        "n": "indexer",
        "N": "limitval"
      },
      "question": "Problem A-4\nFor \\( 0<variable<1 \\), express\n\\[\n\\sum_{indexer=0}^{\\infty} \\frac{variable^{2^{indexer}}}{1-variable^{2^{indexer+1}}}\n\\]\nas a rational function of \\( variable \\).",
      "solution": "A-4.\n\\[\n\\begin{aligned}\n\\sum_{indexer=0}^{limitval} \\frac{variable^{2^{indexer}}}{1-variable^{2^{indexer+1}}} & =\\sum_{indexer=0}^{limitval}\\left(\\frac{1}{1-variable^{2^{indexer}}}-\\frac{1}{1-variable^{2^{indexer+1}}}\\right) \\\\\n& =\\frac{1}{1-variable}-\\frac{1}{1-variable^{2^{limitval+1}}} \\rightarrow \\frac{1}{1-variable}-1=\\frac{variable}{1-variable} \\text { as } limitval \\rightarrow \\infty\n\\end{aligned}\n\\]\nsince \\( |variable|<1 \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "diameter",
        "n": "boulevard",
        "N": "raincloud"
      },
      "question": "Problem A-4\nFor \\( 0<diameter<1 \\), express\n\\[\n\\sum_{boulevard=0}^{\\infty} \\frac{diameter^{2^{boulevard}}}{1-diameter^{2^{boulevard+1}}}\n\\]\nas a rational function of \\( diameter \\).",
      "solution": "A-4.\n\\[\n\\begin{aligned}\n\\sum_{boulevard=0}^{raincloud} \\frac{diameter^{2^{boulevard}}}{1-diameter^{2^{boulevard+1}}} & =\\sum_{boulevard=0}^{raincloud}\\left(\\frac{1}{1-diameter^{2^{boulevard}}}-\\frac{1}{1-diameter^{2^{boulevard+1}}}\\right) \\\\\n& =\\frac{1}{1-diameter}-\\frac{1}{1-diameter^{2^{raincloud+1}}} \\rightarrow \\frac{1}{1-diameter}-1=\\frac{diameter}{1-diameter} \\text { as } raincloud \\rightarrow \\infty\n\\end{aligned}\n\\]\nsince \\( |diameter|<1 \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "hugevalue",
        "n": "continuum",
        "N": "limitless"
      },
      "question": "Problem A-4\nFor \\( 0<hugevalue<1 \\), express\n\\[\n\\sum_{continuum=0}^{\\infty} \\frac{hugevalue^{2^{continuum}}}{1-hugevalue^{2^{continuum+1}}}\n\\]\nas a rational function of \\( hugevalue \\).",
      "solution": "A-4.\n\\[\n\\begin{aligned}\n\\sum_{continuum=0}^{limitless} \\frac{hugevalue^{2^{continuum}}}{1-hugevalue^{2^{continuum+1}}} & =\\sum_{continuum=0}^{limitless}\\left(\\frac{1}{1-hugevalue^{2^{continuum}}}-\\frac{1}{1-hugevalue^{2^{continuum+1}}}\\right) \\\\\n& =\\frac{1}{1-hugevalue}-\\frac{1}{1-hugevalue^{2^{limitless+1}}} \\rightarrow \\frac{1}{1-hugevalue}-1=\\frac{hugevalue}{1-hugevalue} \\text { as } limitless \\rightarrow \\infty\n\\end{aligned}\n\\]\nsince \\( |hugevalue|<1 \\)."
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "n": "hjgrksla",
        "N": "mnbvcxzi"
      },
      "question": "Problem A-4\nFor \\( 0<qzxwvtnp<1 \\), express\n\\[\n\\sum_{hjgrksla=0}^{\\infty} \\frac{qzxwvtnp^{2^{hjgrksla}}}{1-qzxwvtnp^{2^{hjgrksla+1}}}\n\\]\nas a rational function of \\( qzxwvtnp \\).",
      "solution": "A-4.\n\\[\n\\begin{aligned}\n\\sum_{hjgrksla=0}^{mnbvcxzi} \\frac{qzxwvtnp^{2^{hjgrksla}}}{1-qzxwvtnp^{2^{hjgrksla+1}}} & =\\sum_{hjgrksla=0}^{mnbvcxzi}\\left(\\frac{1}{1-qzxwvtnp^{2^{hjgrksla}}}-\\frac{1}{1-qzxwvtnp^{2^{hjgrksla+1}}}\\right) \\\\\n& =\\frac{1}{1-qzxwvtnp}-\\frac{1}{1-qzxwvtnp^{2^{mnbvcxzi+1}}} \\rightarrow \\frac{1}{1-qzxwvtnp}-1=\\frac{qzxwvtnp}{1-qzxwvtnp} \\text { as } mnbvcxzi \\rightarrow \\infty\n\\end{aligned}\n\\]\nsince \\( |qzxwvtnp|<1 \\)."
    },
    "kernel_variant": {
      "question": "Let x be a real number with −1 < x < 1.  Evaluate, in closed form, the infinite series  \n\n\\[\nT(x)=\\sum_{n=0}^{\\infty}\\Biggl(\n\\frac{1}{1-x^{2^{\\,n}}}-4\\frac{1}{1-x^{2^{\\,n+1}}}+6\\frac{1}{1-x^{2^{\\,n+2}}}\n-4\\frac{1}{1-x^{2^{\\,n+3}}}+\\frac{1}{1-x^{2^{\\,n+4}}}\\Biggr),\n\\]\n\nand express the answer as a single rational function of x.",
      "solution": "1.  Introduce the shorthand  \n   \\[\n      A_k=\\frac{1}{1-x^{2^{k}}}\\qquad(k\\ge 0).\n   \\]\n\n2.  Rewrite every summand in terms of the sequence \\((A_k)\\): for fixed n  \n\n   \\[\n   S_n=A_{n}-4A_{n+1}+6A_{n+2}-4A_{n+3}+A_{n+4}.\n   \\]\n\n   Hence  \n\n   \\[\n      T(x)=\\sum_{n=0}^{\\infty}S_n\n          =\\sum_{n=0}^{\\infty}\\sum_{j=0}^{4}(-1)^j\\binom{4}{j}\\,A_{n+j}.\n   \\]\n\n3.  Collect the coefficient of each individual \\(A_k\\) in the double sum.  \n   A given index \\(k\\) appears in those inner terms with \\(n=k,\\;k-1,\\;k-2,\\;k-3,\\;k-4\\); consequently its total coefficient is  \n\n   \\[\n      c_k=\\sum_{j=0}^{4}(-1)^j\\binom{4}{j}\\mathbf 1_{\\{0\\le j\\le k\\}},\n   \\]\n   where \\(\\mathbf 1\\) is the indicator function.  When \\(k\\ge 4\\) the entire binomial sum is present and \\(c_k=0\\) because \\(\\sum_{j=0}^{4}(-1)^j\\binom{4}{j}=0\\).  Thus only the first four indices survive:\n\n   \\[\n      c_0=1,\\qquad c_1=-3,\\qquad c_2=3,\\qquad c_3=-1.\n   \\]\n\n   Therefore  \n\n   \\[\n      T(x)=A_0-3A_1+3A_2-A_3\n           =\\frac1{1-x}-\\frac{3}{1-x^{2}}+\\frac{3}{1-x^{4}}-\\frac{1}{1-x^{8}}.\n   \\]\n\n4.  Put over a common denominator.  Observe  \n\n   \\[\n      (1-x)(1+x)(1+x^{2})(1+x^{4})=1-x^{8}.\n   \\]\n\n   Using  \n   \\[\n      \\frac1{1-x^{2}}=\\frac1{(1-x)(1+x)},\\qquad\n      \\frac1{1-x^{4}}=\\frac1{(1-x)(1+x)(1+x^{2})},\n   \\]\n   and \\(\\displaystyle\\frac1{1-x^{8}}=\\frac1{(1-x)(1+x)(1+x^{2})(1+x^{4})}\\),\n   we write every term with the denominator \\(1-x^{8}\\):\n\n   \\[\n   \\begin{aligned}\n      T(x)&=\\frac{(1+x)(1+x^{2})(1+x^{4})\n              -3(1+x^{2})(1+x^{4})\n              +3(1+x^{4})\n              -1}{1-x^{8}}\\\\[4pt]\n           &=\\frac{x-2x^{2}+x^{3}+x^{4}+x^{5}-2x^{6}+x^{7}}{1-x^{8}}.\n   \\end{aligned}\n   \\]\n\n   Hence  \n\n   \\[\n      \\boxed{T(x)=\\dfrac{x-2x^{2}+x^{3}+x^{4}+x^{5}-2x^{6}+x^{7}}{1-x^{8}}}.\n   \\]\n\n   Because |x|<1, the infinite series converges and the rational expression is well-defined.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.630872",
        "was_fixed": false,
        "difficulty_analysis": "• Higher-order interaction: the original series telescopes after a single forward difference; here we impose a 4-th order finite difference, producing five interacting terms whose cancellations must be tracked carefully.  \n• Binomial-coefficient structure: identifying that the coefficients 1, –4, 6, –4, 1 represent the 4-th forward difference of the sequence \\((A_k)\\) is essential; simple pattern matching no longer suffices.  \n• Boundary analysis: all but the first four indices vanish only after a non-trivial combinatorial count of contributions.  \n• Algebraic complexity: the surviving terms involve four nested products \\((1\\pm x)(1+x^{2})(1+x^{4})\\), demanding meticulous common-denominator work and factor simplifications to reach the compact rational form.  \nTogether these steps add several layers of technical depth beyond the original problem’s single telescoping trick, making the enhanced variant significantly harder."
      }
    },
    "original_kernel_variant": {
      "question": "Let x be a real number with −1 < x < 1.  Evaluate, in closed form, the infinite series  \n\n\\[\nT(x)=\\sum_{n=0}^{\\infty}\\Biggl(\n\\frac{1}{1-x^{2^{\\,n}}}-4\\frac{1}{1-x^{2^{\\,n+1}}}+6\\frac{1}{1-x^{2^{\\,n+2}}}\n-4\\frac{1}{1-x^{2^{\\,n+3}}}+\\frac{1}{1-x^{2^{\\,n+4}}}\\Biggr),\n\\]\n\nand express the answer as a single rational function of x.",
      "solution": "1.  Introduce the shorthand  \n   \\[\n      A_k=\\frac{1}{1-x^{2^{k}}}\\qquad(k\\ge 0).\n   \\]\n\n2.  Rewrite every summand in terms of the sequence \\((A_k)\\): for fixed n  \n\n   \\[\n   S_n=A_{n}-4A_{n+1}+6A_{n+2}-4A_{n+3}+A_{n+4}.\n   \\]\n\n   Hence  \n\n   \\[\n      T(x)=\\sum_{n=0}^{\\infty}S_n\n          =\\sum_{n=0}^{\\infty}\\sum_{j=0}^{4}(-1)^j\\binom{4}{j}\\,A_{n+j}.\n   \\]\n\n3.  Collect the coefficient of each individual \\(A_k\\) in the double sum.  \n   A given index \\(k\\) appears in those inner terms with \\(n=k,\\;k-1,\\;k-2,\\;k-3,\\;k-4\\); consequently its total coefficient is  \n\n   \\[\n      c_k=\\sum_{j=0}^{4}(-1)^j\\binom{4}{j}\\mathbf 1_{\\{0\\le j\\le k\\}},\n   \\]\n   where \\(\\mathbf 1\\) is the indicator function.  When \\(k\\ge 4\\) the entire binomial sum is present and \\(c_k=0\\) because \\(\\sum_{j=0}^{4}(-1)^j\\binom{4}{j}=0\\).  Thus only the first four indices survive:\n\n   \\[\n      c_0=1,\\qquad c_1=-3,\\qquad c_2=3,\\qquad c_3=-1.\n   \\]\n\n   Therefore  \n\n   \\[\n      T(x)=A_0-3A_1+3A_2-A_3\n           =\\frac1{1-x}-\\frac{3}{1-x^{2}}+\\frac{3}{1-x^{4}}-\\frac{1}{1-x^{8}}.\n   \\]\n\n4.  Put over a common denominator.  Observe  \n\n   \\[\n      (1-x)(1+x)(1+x^{2})(1+x^{4})=1-x^{8}.\n   \\]\n\n   Using  \n   \\[\n      \\frac1{1-x^{2}}=\\frac1{(1-x)(1+x)},\\qquad\n      \\frac1{1-x^{4}}=\\frac1{(1-x)(1+x)(1+x^{2})},\n   \\]\n   and \\(\\displaystyle\\frac1{1-x^{8}}=\\frac1{(1-x)(1+x)(1+x^{2})(1+x^{4})}\\),\n   we write every term with the denominator \\(1-x^{8}\\):\n\n   \\[\n   \\begin{aligned}\n      T(x)&=\\frac{(1+x)(1+x^{2})(1+x^{4})\n              -3(1+x^{2})(1+x^{4})\n              +3(1+x^{4})\n              -1}{1-x^{8}}\\\\[4pt]\n           &=\\frac{x-2x^{2}+x^{3}+x^{4}+x^{5}-2x^{6}+x^{7}}{1-x^{8}}.\n   \\end{aligned}\n   \\]\n\n   Hence  \n\n   \\[\n      \\boxed{T(x)=\\dfrac{x-2x^{2}+x^{3}+x^{4}+x^{5}-2x^{6}+x^{7}}{1-x^{8}}}.\n   \\]\n\n   Because |x|<1, the infinite series converges and the rational expression is well-defined.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.503180",
        "was_fixed": false,
        "difficulty_analysis": "• Higher-order interaction: the original series telescopes after a single forward difference; here we impose a 4-th order finite difference, producing five interacting terms whose cancellations must be tracked carefully.  \n• Binomial-coefficient structure: identifying that the coefficients 1, –4, 6, –4, 1 represent the 4-th forward difference of the sequence \\((A_k)\\) is essential; simple pattern matching no longer suffices.  \n• Boundary analysis: all but the first four indices vanish only after a non-trivial combinatorial count of contributions.  \n• Algebraic complexity: the surviving terms involve four nested products \\((1\\pm x)(1+x^{2})(1+x^{4})\\), demanding meticulous common-denominator work and factor simplifications to reach the compact rational form.  \nTogether these steps add several layers of technical depth beyond the original problem’s single telescoping trick, making the enhanced variant significantly harder."
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}