{
  "index": "1970-B-1",
  "type": "ANA",
  "tag": [
    "ANA",
    "ALG"
  ],
  "difficulty": "",
  "question": "\\begin{array}{l}\n\\text { B-1. Evaluate }\\\\\n\\operatorname{limit}_{n \\rightarrow \\infty} \\frac{1}{n^{4}} \\prod_{i=1}^{2 n}\\left(n^{2}+i^{2}\\right)^{1 / n}\n\\end{array}",
  "solution": "B-1\nLet\n\\[\na_{n}=\\frac{1}{n^{4}} \\prod_{i=1}^{2 n}\\left(n^{2}+i^{2}\\right)^{1 / n}\n\\]\n\nThen\n\\[\n\\log a_{n}=\\frac{1}{n} \\sum_{i=1}^{2 n} \\log \\left(1+\\frac{i^{2}}{n^{2}}\\right)\n\\]\nand\n\\[\n\\operatorname{limit}_{n \\rightarrow \\infty} \\log a_{n}=\\int_{0}^{2} \\log \\left(1+x^{2}\\right) d x=2 \\log 5-4+2 \\arctan 2\n\\]",
  "vars": [
    "n",
    "i",
    "a_n",
    "x"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "n": "bigcount",
        "i": "inneridx",
        "a_n": "sequence",
        "x": "dummyvar"
      },
      "question": "\\begin{array}{l}\n\\text { B-1. Evaluate }\\\\\n\\operatorname{limit}_{bigcount \\rightarrow \\infty} \\frac{1}{bigcount^{4}} \\prod_{inneridx=1}^{2 bigcount}\\left(bigcount^{2}+inneridx^{2}\\right)^{1 / bigcount}\n\\end{array}",
      "solution": "B-1\nLet\n\\[\nsequence=\\frac{1}{bigcount^{4}} \\prod_{inneridx=1}^{2 bigcount}\\left(bigcount^{2}+inneridx^{2}\\right)^{1 / bigcount}\n\\]\n\nThen\n\\[\n\\log sequence=\\frac{1}{bigcount} \\sum_{inneridx=1}^{2 bigcount} \\log \\left(1+\\frac{inneridx^{2}}{bigcount^{2}}\\right)\n\\]\nand\n\\[\n\\operatorname{limit}_{bigcount \\rightarrow \\infty} \\log sequence=\\int_{0}^{2} \\log \\left(1+dummyvar^{2}\\right) d dummyvar=2 \\log 5-4+2 \\arctan 2\n\\]"
    },
    "descriptive_long_confusing": {
      "map": {
        "n": "longitude",
        "a_n": "marigold",
        "x": "moonlight"
      },
      "question": "\\begin{array}{l}\n\\text { B-1. Evaluate }\\\\\n\\operatorname{limit}_{longitude \\rightarrow \\infty} \\frac{1}{longitude^{4}} \\prod_{i=1}^{2 longitude}\\left(longitude^{2}+i^{2}\\right)^{1 / longitude}\n\\end{array}",
      "solution": "B-1\nLet\n\\[\nmarigold=\\frac{1}{longitude^{4}} \\prod_{i=1}^{2 longitude}\\left(longitude^{2}+i^{2}\\right)^{1 / longitude}\n\\]\n\nThen\n\\[\n\\log marigold=\\frac{1}{longitude} \\sum_{i=1}^{2 longitude} \\log \\left(1+\\frac{i^{2}}{longitude^{2}}\\right)\n\\]\nand\n\\[\n\\operatorname{limit}_{longitude \\rightarrow \\infty} \\log marigold=\\int_{0}^{2} \\log \\left(1+moonlight^{2}\\right) d moonlight=2 \\log 5-4+2 \\arctan 2\n\\]\n"
    },
    "descriptive_long_misleading": {
      "map": {
        "n": "finitevar",
        "i": "wholevalue",
        "a_n": "constantvalue",
        "x": "infinitevar"
      },
      "question": "\\begin{array}{l}\n\\text { B-1. Evaluate }\\\\\n\\operatorname{limit}_{finitevar \\rightarrow \\infty} \\frac{1}{finitevar^{4}} \\prod_{wholevalue=1}^{2 finitevar}\\left(finitevar^{2}+wholevalue^{2}\\right)^{1 / finitevar}\n\\end{array}",
      "solution": "B-1\nLet\n\\[\nconstantvalue=\\frac{1}{finitevar^{4}} \\prod_{wholevalue=1}^{2 finitevar}\\left(finitevar^{2}+wholevalue^{2}\\right)^{1 / finitevar}\n\\]\n\nThen\n\\[\n\\log constantvalue=\\frac{1}{finitevar} \\sum_{wholevalue=1}^{2 finitevar} \\log \\left(1+\\frac{wholevalue^{2}}{finitevar^{2}}\\right)\n\\]\nand\n\\[\n\\operatorname{limit}_{finitevar \\rightarrow \\infty} \\log constantvalue=\\int_{0}^{2} \\log \\left(1+infinitevar^{2}\\right) d infinitevar=2 \\log 5-4+2 \\arctan 2\n\\]"
    },
    "garbled_string": {
      "map": {
        "n": "qzxwvtnp",
        "i": "hjgrksla",
        "a_n": "vbcksdfo",
        "x": "zlkmnwyq"
      },
      "question": "\\begin{array}{l}\n\\text { B-1. Evaluate }\\\\\n\\operatorname{limit}_{qzxwvtnp \\rightarrow \\infty} \\frac{1}{qzxwvtnp^{4}} \\prod_{hjgrksla=1}^{2 qzxwvtnp}\\left(qzxwvtnp^{2}+hjgrksla^{2}\\right)^{1 / qzxwvtnp}\n\\end{array}",
      "solution": "B-1\nLet\n\\[\nvbcksdfo=\\frac{1}{qzxwvtnp^{4}} \\prod_{hjgrksla=1}^{2 qzxwvtnp}\\left(qzxwvtnp^{2}+hjgrksla^{2}\\right)^{1 / qzxwvtnp}\n\\]\n\nThen\n\\[\n\\log vbcksdfo=\\frac{1}{qzxwvtnp} \\sum_{hjgrksla=1}^{2 qzxwvtnp} \\log \\left(1+\\frac{hjgrksla^{2}}{qzxwvtnp^{2}}\\right)\n\\]\nand\n\\[\n\\operatorname{limit}_{qzxwvtnp \\rightarrow \\infty} \\log vbcksdfo=\\int_{0}^{2} \\log \\left(1+zlkmnwyq^{2}\\right) d zlkmnwyq=2 \\log 5-4+2 \\arctan 2\n\\]"
    },
    "kernel_variant": {
      "question": "Evaluate\n\\[\n\\lim_{n\\to\\infty} \\frac{1}{n^{12}} \\prod_{i=1}^{4n} \\left(n + 3 i\\right)^{3/n} .\n\\]",
      "solution": "Set\n\\[\na_n=\\frac{1}{n^{12}}\\prod_{i=1}^{4n}(n+3i)^{3/n}.\n\\]\n1. Take natural logarithms:\n\\[\n\\log a_n=\\frac{3}{n}\\sum_{i=1}^{4n}\\log(n+3i)-12\\log n.\n\\]\n2. Factor out n:\n\\[\n\\log(n+3i)=\\log n+\\log\\Bigl(1+\\tfrac{3i}{n}\\Bigr),\n\\]\nso\n\\[\n\\log a_n=\\frac{3}{n}\\sum_{i=1}^{4n}\\log n+\\frac{3}{n}\\sum_{i=1}^{4n}\\log\\Bigl(1+\\tfrac{3i}{n}\\Bigr)-12\\log n.\n\\]\nThe first summation is $(3/n)(4n)\\log n=12\\log n$, which cancels $-12\\log n$.  Thus\n\\[\n\\log a_n=\\frac{3}{n}\\sum_{i=1}^{4n}\\log\\Bigl(1+\\tfrac{3i}{n}\\Bigr).\n\\]\n3. Recognize a Riemann sum.  With $x_i=i/n$,\n\\[\n\\frac{1}{n}\\sum_{i=1}^{4n}\\log(1+3x_i)\\longrightarrow\\int_{0}^{4}\\log(1+3x)\\,dx\n\\]\nas $n\\to\\infty$.  Multiplying by 3 gives\n\\[\n\\lim_{n\\to\\infty}\\log a_n=3\\int_{0}^{4}\\log(1+3x)\\,dx.\n\\]\n4. Evaluate the integral via $u=1+3x$, $du=3dx$:\n\\[\n\\int_{0}^{4}\\log(1+3x)\\,dx=\\frac13\\int_{1}^{13}\\log u\\,du=\\frac13\\bigl[u\\log u-u\\bigr]_{1}^{13}=\\frac13\\bigl(13\\log13-12\\bigr).\n\\]\nHence\n\\[\n\\lim_{n\\to\\infty}\\log a_n=3\\cdot\\tfrac13\\bigl(13\\log13-12\\bigr)=13\\log13-12,\n\\]\nand exponentiating gives\n\\[\n\\lim_{n\\to\\infty}\\frac{1}{n^{12}}\\prod_{i=1}^{4n}(n+3i)^{3/n}=\\frac{13^{13}}{e^{12}}.\n\\]",
      "_meta": {
        "core_steps": [
          "Apply the natural logarithm to turn the product into a sum.",
          "Factor out n^q from each term so every summand becomes log(1 + a·(i/n)^q).",
          "Recognize (k/n)·Σ_{i=1}^{c n} f(i/n) as a Riemann sum → ∫_{0}^{c} log(1 + a x^q) dx.",
          "Evaluate the integral and exponentiate to recover the desired limit."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Constant multiple c determining how far the index i runs (upper limit c·n and hence the integration interval [0, c]).",
            "original": 2
          },
          "slot2": {
            "description": "Common power q applied to both n and i inside the sum-and-integrand: n^q + a·i^q.",
            "original": 2
          },
          "slot3": {
            "description": "Constant coefficient a on the i^q term (gives log(1 + a·(i/n)^q)).",
            "original": 1
          },
          "slot4": {
            "description": "Factor k in the microscopic exponent k/n on each term of the product.",
            "original": 1
          },
          "slot5": {
            "description": "Macroscopic power s = q·c·k on the prefactor n^{s} that must be cancelled outside (here shown as n^{4}).",
            "original": 4
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}