{
  "index": "1960-B-6",
  "type": "ANA",
  "tag": [
    "ANA",
    "NT"
  ],
  "difficulty": "",
  "question": "6. Any positive integer may be written in the form \\( n=2^{k}(2 l+1) \\). Let \\( a_{n} \\) \\( =e^{-k} \\) and \\( b_{n}=a_{1} a_{2} a_{3} \\cdots a_{n} \\). Prove that \\( \\Sigma b_{n} \\) converges.",
  "solution": "Solution. It is clear that \\( a_{n}=e^{0}=1 \\) if \\( n \\) is odd and \\( a_{n} \\leq e^{-1} \\) if \\( n \\) is even. Therefore\n\\[\nb_{2 k}=a_{1} a_{2} \\cdots a_{2 k} \\leq e^{-k},\n\\]\nand\n\\[\nb_{2 k+1} \\leq e^{-k}\n\\]\n\nTherefore,\n\\[\n\\begin{aligned}\nb_{1}+b_{2}+\\cdots b_{2 k} & <b_{1}+b_{2}+\\cdots+b_{2 k+1} \\\\\n& \\leq 1+2 e^{-1}+2 e^{-2}+\\cdots+2 e^{-2 k}<1+\\frac{2 e^{-1}}{1-e^{-1}}\n\\end{aligned}\n\\]\n\nThus the partial sums of \\( \\Sigma b_{n} \\) are bounded. Since the series has positive terms, it converges.",
  "vars": [
    "n",
    "k",
    "l",
    "a_n",
    "b_n",
    "a_1",
    "a_2",
    "a_3",
    "b_2k",
    "b_2k+1"
  ],
  "params": [],
  "sci_consts": [
    "e"
  ],
  "variants": {
    "descriptive_long": {
      "map": {
        "n": "posintnum",
        "k": "twopowerexp",
        "l": "oddpartpar",
        "a_n": "aseqelem",
        "b_n": "bseqelem",
        "a_1": "aseqfirst",
        "a_2": "aseqsecond",
        "a_3": "aseqthird",
        "b_2k": "bseqeven",
        "b_2k+1": "bseqoddnext"
      },
      "question": "6. Any positive integer may be written in the form \\( posintnum=2^{twopowerexp}(2 oddpartpar+1) \\). Let \\( aseqelem \\) \\( =e^{-twopowerexp} \\) and \\( bseqelem=aseqfirst aseqsecond aseqthird \\cdots aseqelem \\). Prove that \\( \\Sigma bseqelem \\) converges.",
      "solution": "Solution. It is clear that \\( aseqelem=e^{0}=1 \\) if \\( posintnum \\) is odd and \\( aseqelem \\leq e^{-1} \\) if \\( posintnum \\) is even. Therefore\n\\[\nbseqeven=a_{1} a_{2} \\cdots a_{2\\,twopowerexp} \\leq e^{-twopowerexp},\n\\]\nand\n\\[\nbseqoddnext \\leq e^{-twopowerexp}\n\\]\nTherefore,\n\\[\n\\begin{aligned}\nb_{1}+b_{2}+\\cdots bseqeven & <b_{1}+b_{2}+\\cdots+bseqoddnext \\\\\n& \\leq 1+2 e^{-1}+2 e^{-2}+\\cdots+2 e^{-2\\,twopowerexp}<1+\\frac{2 e^{-1}}{1-e^{-1}}\n\\end{aligned}\n\\]\nThus the partial sums of \\( \\Sigma bseqelem \\) are bounded. Since the series has positive terms, it converges."
    },
    "descriptive_long_confusing": {
      "map": {
        "n": "timberwolf",
        "k": "marigolds",
        "l": "sandpiper",
        "a_n": "cranberries",
        "b_n": "cornstarch",
        "a_1": "blacksmith",
        "a_2": "toothbrush",
        "a_3": "peppercorn",
        "b_2k": "raincloud",
        "b_2k+1": "dragonfly"
      },
      "question": "Any positive integer may be written in the form \\( timberwolf=2^{marigolds}(2 sandpiper+1) \\). Let \\( cranberries=e^{-marigolds} \\) and \\( cornstarch=blacksmith toothbrush peppercorn \\cdots cranberries \\). Prove that \\( \\Sigma cornstarch \\) converges.",
      "solution": "Solution. It is clear that \\( cranberries=e^{0}=1 \\) if \\( timberwolf \\) is odd and \\( cranberries \\leq e^{-1} \\) if \\( timberwolf \\) is even. Therefore\n\\[\nraincloud=blacksmith toothbrush \\cdots a_{2 marigolds} \\leq e^{-marigolds},\n\\]\nand\n\\[\ndragonfly \\leq e^{-marigolds}\n\\]\n\nTherefore,\n\\[\n\\begin{aligned}\n b_{1}+b_{2}+\\cdots raincloud & <b_{1}+b_{2}+\\cdots+dragonfly \\\\\n & \\leq 1+2 e^{-1}+2 e^{-2}+\\cdots+2 e^{-2 marigolds}<1+\\frac{2 e^{-1}}{1-e^{-1}}\n\\end{aligned}\n\\]\n\nThus the partial sums of \\( \\Sigma cornstarch \\) are bounded. Since the series has positive terms, it converges."
    },
    "descriptive_long_misleading": {
      "map": {
        "n": "fractionvalue",
        "k": "logarithm",
        "l": "negative",
        "a_n": "largeseries",
        "b_n": "quotientseq",
        "a_1": "giganticone",
        "a_2": "gigantictwo",
        "a_3": "giganticthree",
        "b_2k": "quotienteven",
        "b_2k+1": "quotientodd"
      },
      "question": "<<<\n6. Any positive integer may be written in the form \\( fractionvalue=2^{logarithm}(2 negative+1) \\). Let \\( largeseries =e^{-logarithm} \\) and \\( quotientseq=giganticone gigantictwo giganticthree \\cdots largeseries \\). Prove that \\( \\Sigma quotientseq \\) converges.\n>>>",
      "solution": "<<<\nSolution. It is clear that \\( largeseries=e^{0}=1 \\) if \\( fractionvalue \\) is odd and \\( largeseries \\leq e^{-1} \\) if \\( fractionvalue \\) is even. Therefore\n\\[\nquotienteven=giganticone gigantictwo \\cdots largeseries \\leq e^{-logarithm},\n\\]\nand\n\\[\nquotientodd \\leq e^{-logarithm}\n\\]\nTherefore,\n\\[\n\\begin{aligned}\nb_{1}+b_{2}+\\cdots+quotienteven & <b_{1}+b_{2}+\\cdots+quotientodd \\\\\n& \\leq 1+2 e^{-1}+2 e^{-2}+\\cdots+2 e^{-2 logarithm}<1+\\frac{2 e^{-1}}{1-e^{-1}}\n\\end{aligned}\n\\]\nThus the partial sums of \\( \\Sigma quotientseq \\) are bounded. Since the series has positive terms, it converges.\n>>>"
    },
    "garbled_string": {
      "map": {
        "n": "qzxwvtnp",
        "k": "hjgrksla",
        "l": "mdfqplzo",
        "a_n": "rcptebha",
        "b_n": "kvusymni",
        "a_1": "sblqtrwo",
        "a_2": "pfkxajcm",
        "a_3": "yvnhqzig",
        "b_2k": "wdrxoful",
        "b_2k+1": "tnmqzsky"
      },
      "question": "6. Any positive integer may be written in the form \\( qzxwvtnp=2^{hjgrksla}(2 mdfqplzo+1) \\). Let \\( rcptebha =e^{-hjgrksla} \\) and \\( kvusymni=sblqtrwo pfkxajcm yvnhqzig \\cdots rcptebha \\). Prove that \\( \\Sigma kvusymni \\) converges.",
      "solution": "Solution. It is clear that \\( rcptebha=e^{0}=1 \\) if \\( qzxwvtnp \\) is odd and \\( rcptebha \\leq e^{-1} \\) if \\( qzxwvtnp \\) is even. Therefore\n\\[\nwdrxoful=sblqtrwo pfkxajcm \\cdots a_{2 hjgrksla} \\leq e^{-hjgrksla},\n\\]\nand\n\\[\ntnmqzsky \\leq e^{-hjgrksla}\n\\]\nTherefore,\n\\[\n\\begin{aligned}\nb_{1}+b_{2}+\\cdots wdrxoful & <b_{1}+b_{2}+\\cdots+tnmqzsky \\\\\n& \\leq 1+2 e^{-1}+2 e^{-2}+\\cdots+2 e^{-2 hjgrksla}<1+\\frac{2 e^{-1}}{1-e^{-1}}\n\\end{aligned}\n\\]\nThus the partial sums of \\( \\Sigma kvusymni \\) are bounded. Since the series has positive terms, it converges."
    },
    "kernel_variant": {
      "question": "Every positive integer admits a unique factorisation  \n n = 2^{α(n)} 3^{β(n)} m with gcd(m, 6)=1.  \nDefine a_n = 7^{-α(n)} 5^{-β(n)} and b_n = ∏_{j=1}^{n} a_j.  \nProve that the series ∑_{n=1}^{∞} b_n converges and give an explicit numerical upper bound for its sum.\n\n────────────────────────",
      "solution": "(≈ 60 words, original style)  \nWrite n as above and set a_n, b_n accordingly.  \nNote that a_n=1 if 6∤n; a_n≤7^{-1} when 2|n, 3∤n; a_n≤5^{-1} when 3|n, 2∤n; a_n≤35^{-1} when 6|n.  \nAmong 1,…,n there are ⌊n/2⌋ even numbers and ⌊n/3⌋ multiples of 3, whence  \n\n b_n ≤ 7^{-⌊n/2⌋} 5^{-⌊n/3⌋} ≤ 35·(35)^{-n/6}.  \n\nThus ∑ b_n ≤ 35 ∑_{n≥1}(35)^{-n/6}=35·(35^{-1/6})/(1−35^{-1/6})<∞, so the series converges.\n\n────────────────────────",
      "_replacement_note": {
        "replaced_at": "2025-07-05T22:17:12.019664",
        "reason": "Original kernel variant was too easy compared to the original problem"
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}