{
  "index": "1960-B-2",
  "type": "ANA",
  "tag": [
    "ANA",
    "NT"
  ],
  "difficulty": "",
  "question": "\\begin{array}{l}\n\\text { 2. Evaluate the double series }\\\\\n\\sum_{j=0}^{\\infty} \\sum_{k=0}^{\\infty} 2^{-3 k-j-(k+1)^{2}} .\n\\end{array}",
  "solution": "Solution. If we write out a few terms of the double sequence of terms, we see that each of the terms \\( 2^{-2 m} \\) occurs exactly once. Since all the terms\n\\begin{tabular}{l|llll}\n\\( j \\) & 0 & 1 & 2 & 3 \\\\\n\\cline { 2 - 5 } & \\( 2^{0} \\) & \\( 2^{-4} \\) & \\( 2^{-10} \\) & \\( 2^{-18} \\) \\\\\n1 & \\( 2^{-2} \\) & \\( 2^{-8} \\) & \\( 2^{-16} \\) & \\\\\n2 & \\( 2^{-6} \\) & \\( 2^{-14} \\) & & \\\\\n3 & \\( 2^{-12} \\) & & &\n\\end{tabular}\nare positive, the double series may be rearranged to make the simple series\n\\[\n\\sum_{m=0}^{\\infty} 2^{-2 m}\n\\]\nwhich sums to \\( 4 / 3 \\). The sum of the original double series is therefore \\( 4 / 3 \\). To formalize this argument one must prove that\n\\[\n(j, k) \\mapsto(k+j)^{2}+3 k+j\n\\]\nis a bijection from the set of ordered pairs of non-negative integers to the set of non-negative even integers. This is straightforward but long.\n\nA direct analytic argument can be given as follows: Sum the double series along the diagonals defined by \\( j+k=n \\), and then sum on \\( n \\).\n\nWe find\n\\[\n\\begin{aligned}\n\\sum_{j=0}^{\\infty} \\sum_{k=0}^{\\infty} 2^{-3 k-j-(j+k)^{2}} & =\\sum_{n=0}^{\\infty} \\sum_{k=0}^{n} 2^{-2 k-n-n^{2}} \\\\\n& =\\sum_{n=0}^{\\infty}\\left[\\frac{4}{3}\\left(1-2^{-2 n-2}\\right) 2^{-n-n^{2}}\\right] \\\\\n& =\\frac{4}{3}\\left[\\sum_{n=0}^{\\infty} 2^{-n-n^{2}}-\\sum_{n=0}^{\\infty} 2^{-(n+1)(n+2)}\\right] \\\\\n& =\\frac{4}{3}\\left[\\sum_{n=0}^{\\infty} 2^{-n(n+1)}-\\sum_{m=1}^{\\infty} 2^{-m(m+1)}\\right] \\\\\n& =4 / 3\n\\end{aligned}\n\\]\n\nAt the third step, the separation of the sum into two sums is permissible because the new sums are convergent.",
  "vars": [
    "j",
    "k",
    "m",
    "n"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "j": "indexalpha",
        "k": "indexbeta",
        "m": "indexgamma",
        "n": "indexdelta"
      },
      "question": "\\begin{array}{l}\n\\text { 2. Evaluate the double series }\\\\\n\\sum_{indexalpha=0}^{\\infty} \\sum_{indexbeta=0}^{\\infty} 2^{-3 indexbeta-indexalpha-(indexbeta+1)^{2}} .\n\\end{array}",
      "solution": "Solution. If we write out a few terms of the double sequence of terms, we see that each of the terms \\( 2^{-2 indexgamma} \\) occurs exactly once. Since all the terms\n\\begin{tabular}{l|llll}\n\\( indexalpha \\) & 0 & 1 & 2 & 3 \\\\\n\\cline { 2 - 5 } & \\( 2^{0} \\) & \\( 2^{-4} \\) & \\( 2^{-10} \\) & \\( 2^{-18} \\) \\\\\n1 & \\( 2^{-2} \\) & \\( 2^{-8} \\) & \\( 2^{-16} \\) & \\\\\n2 & \\( 2^{-6} \\) & \\( 2^{-14} \\) & & \\\\\n3 & \\( 2^{-12} \\) & & &\n\\end{tabular}\nare positive, the double series may be rearranged to make the simple series\n\\[\n\\sum_{indexgamma=0}^{\\infty} 2^{-2 indexgamma}\n\\]\nwhich sums to \\( 4 / 3 \\). The sum of the original double series is therefore \\( 4 / 3 \\). To formalize this argument one must prove that\n\\[\n(indexalpha, indexbeta) \\mapsto(indexbeta+indexalpha)^{2}+3 indexbeta+indexalpha\n\\]\nis a bijection from the set of ordered pairs of non-negative integers to the set of non-negative even integers. This is straightforward but long.\n\nA direct analytic argument can be given as follows: Sum the double series along the diagonals defined by \\( indexalpha+indexbeta=indexdelta \\), and then sum on \\( indexdelta \\).\n\nWe find\n\\[\n\\begin{aligned}\n\\sum_{indexalpha=0}^{\\infty} \\sum_{indexbeta=0}^{\\infty} 2^{-3 indexbeta-indexalpha-(indexalpha+indexbeta)^{2}} & =\\sum_{indexdelta=0}^{\\infty} \\sum_{indexbeta=0}^{indexdelta} 2^{-2 indexbeta-indexdelta-indexdelta^{2}} \\\\\n& =\\sum_{indexdelta=0}^{\\infty}\\left[\\frac{4}{3}\\left(1-2^{-2 indexdelta-2}\\right) 2^{-indexdelta-indexdelta^{2}}\\right] \\\\\n& =\\frac{4}{3}\\left[\\sum_{indexdelta=0}^{\\infty} 2^{-indexdelta-indexdelta^{2}}-\\sum_{indexdelta=0}^{\\infty} 2^{-(indexdelta+1)(indexdelta+2)}\\right] \\\\\n& =\\frac{4}{3}\\left[\\sum_{indexdelta=0}^{\\infty} 2^{-indexdelta(indexdelta+1)}-\\sum_{indexgamma=1}^{\\infty} 2^{-indexgamma(indexgamma+1)}\\right] \\\\\n& =4 / 3\n\\end{aligned}\n\\]\n\nAt the third step, the separation of the sum into two sums is permissible because the new sums are convergent."
    },
    "descriptive_long_confusing": {
      "map": {
        "j": "candlestick",
        "k": "wallpaper",
        "m": "blueberry",
        "n": "caribou"
      },
      "question": "<<<\n\\begin{array}{l}\n\\text { 2. Evaluate the double series }\\\\\n\\sum_{candlestick=0}^{\\infty} \\sum_{wallpaper=0}^{\\infty} 2^{-3 wallpaper-candlestick-(wallpaper+1)^{2}} .\n\\end{array}\n>>>",
      "solution": "<<<\nSolution. If we write out a few terms of the double sequence of terms, we see that each of the terms \\( 2^{-2 blueberry} \\) occurs exactly once. Since all the terms\n\\begin{tabular}{l|llll}\n\\( candlestick \\) & 0 & 1 & 2 & 3 \\\\\n\\cline { 2 - 5 } & \\( 2^{0} \\) & \\( 2^{-4} \\) & \\( 2^{-10} \\) & \\( 2^{-18} \\) \\\\\n1 & \\( 2^{-2} \\) & \\( 2^{-8} \\) & \\( 2^{-16} \\) & \\\\\n2 & \\( 2^{-6} \\) & \\( 2^{-14} \\) & & \\\\\n3 & \\( 2^{-12} \\) & & &\n\\end{tabular}\nare positive, the double series may be rearranged to make the simple series\n\\[\n\\sum_{blueberry=0}^{\\infty} 2^{-2 blueberry}\n\\]\nwhich sums to \\( 4 / 3 \\). The sum of the original double series is therefore \\( 4 / 3 \\). To formalize this argument one must prove that\n\\[\n(candlestick, wallpaper) \\mapsto(wallpaper+candlestick)^{2}+3 wallpaper+candlestick\n\\]\nis a bijection from the set of ordered pairs of non-negative integers to the set of non-negative even integers. This is straightforward but long.\n\nA direct analytic argument can be given as follows: Sum the double series along the diagonals defined by \\( candlestick+wallpaper=caribou \\), and then sum on \\( caribou \\).\n\nWe find\n\\[\n\\begin{aligned}\n\\sum_{candlestick=0}^{\\infty} \\sum_{wallpaper=0}^{\\infty} 2^{-3 wallpaper-candlestick-(candlestick+wallpaper)^{2}} & =\\sum_{caribou=0}^{\\infty} \\sum_{wallpaper=0}^{caribou} 2^{-2 wallpaper-caribou-caribou^{2}} \\\\\n& =\\sum_{caribou=0}^{\\infty}\\left[\\frac{4}{3}\\left(1-2^{-2 caribou-2}\\right) 2^{-caribou-caribou^{2}}\\right] \\\\\n& =\\frac{4}{3}\\left[\\sum_{caribou=0}^{\\infty} 2^{-caribou-caribou^{2}}-\\sum_{caribou=0}^{\\infty} 2^{-(caribou+1)(caribou+2)}\\right] \\\\\n& =\\frac{4}{3}\\left[\\sum_{caribou=0}^{\\infty} 2^{-caribou(caribou+1)}-\\sum_{blueberry=1}^{\\infty} 2^{-blueberry(blueberry+1)}\\right] \\\\\n& =4 / 3\n\\end{aligned}\n\\]\n\nAt the third step, the separation of the sum into two sums is permissible because the new sums are convergent.\n>>>"
    },
    "descriptive_long_misleading": {
      "map": {
        "j": "universal",
        "k": "perpetual",
        "m": "limitless",
        "n": "totality"
      },
      "question": "\\begin{array}{l}\n\\text { 2. Evaluate the double series }\\\\\n\\sum_{universal=0}^{\\infty} \\sum_{perpetual=0}^{\\infty} 2^{-3 perpetual-universal-(perpetual+1)^{2}} .\n\\end{array}",
      "solution": "Solution. If we write out a few terms of the double sequence of terms, we see that each of the terms \\( 2^{-2 limitless} \\) occurs exactly once. Since all the terms\n\\begin{tabular}{l|llll}\n\\( universal \\) & 0 & 1 & 2 & 3 \\\\\n\\cline { 2 - 5 } & \\( 2^{0} \\) & \\( 2^{-4} \\) & \\( 2^{-10} \\) & \\( 2^{-18} \\) \\\\\n1 & \\( 2^{-2} \\) & \\( 2^{-8} \\) & \\( 2^{-16} \\) & \\\\\n2 & \\( 2^{-6} \\) & \\( 2^{-14} \\) & & \\\\\n3 & \\( 2^{-12} \\) & & &\n\\end{tabular}\nare positive, the double series may be rearranged to make the simple series\n\\[\n\\sum_{limitless=0}^{\\infty} 2^{-2 limitless}\n\\]\nwhich sums to \\( 4 / 3 \\). The sum of the original double series is therefore \\( 4 / 3 \\). To formalize this argument one must prove that\n\\[\n(universal, perpetual) \\mapsto(perpetual+universal)^{2}+3 perpetual+universal\n\\]\nis a bijection from the set of ordered pairs of non-negative integers to the set of non-negative even integers. This is straightforward but long.\n\nA direct analytic argument can be given as follows: Sum the double series along the diagonals defined by \\( universal+perpetual=totality \\), and then sum on \\( totality \\).\n\nWe find\n\\[\n\\begin{aligned}\n\\sum_{universal=0}^{\\infty} \\sum_{perpetual=0}^{\\infty} 2^{-3 perpetual-universal-(universal+perpetual)^{2}} & =\\sum_{totality=0}^{\\infty} \\sum_{perpetual=0}^{totality} 2^{-2 perpetual-totality-totality^{2}} \\\\\n& =\\sum_{totality=0}^{\\infty}\\left[\\frac{4}{3}\\left(1-2^{-2 totality-2}\\right) 2^{-totality-totality^{2}}\\right] \\\\\n& =\\frac{4}{3}\\left[\\sum_{totality=0}^{\\infty} 2^{-totality-totality^{2}}-\\sum_{totality=0}^{\\infty} 2^{-(totality+1)(totality+2)}\\right] \\\\\n& =\\frac{4}{3}\\left[\\sum_{totality=0}^{\\infty} 2^{-totality(totality+1)}-\\sum_{limitless=1}^{\\infty} 2^{-limitless(limitless+1)}\\right] \\\\\n& =4 / 3\n\\end{aligned}\n\\]\n\nAt the third step, the separation of the sum into two sums is permissible because the new sums are convergent."
    },
    "garbled_string": {
      "map": {
        "j": "qzxwvtnp",
        "k": "hjgrksla",
        "m": "xptldfou",
        "n": "vrbqmsze"
      },
      "question": "\\begin{array}{l}\n\\text { 2. Evaluate the double series }\\\\\n\\sum_{qzxwvtnp=0}^{\\infty} \\sum_{hjgrksla=0}^{\\infty} 2^{-3 hjgrksla-qzxwvtnp-(hjgrksla+1)^{2}} .\n\\end{array}",
      "solution": "Solution. If we write out a few terms of the double sequence of terms, we see that each of the terms \\( 2^{-2 xptldfou} \\) occurs exactly once. Since all the terms\n\\begin{tabular}{l|llll}\n\\( qzxwvtnp \\) & 0 & 1 & 2 & 3 \\\\\n\\cline { 2 - 5 } & \\( 2^{0} \\) & \\( 2^{-4} \\) & \\( 2^{-10} \\) & \\( 2^{-18} \\) \\\\\n1 & \\( 2^{-2} \\) & \\( 2^{-8} \\) & \\( 2^{-16} \\) & \\\\\n2 & \\( 2^{-6} \\) & \\( 2^{-14} \\) & & \\\\\n3 & \\( 2^{-12} \\) & & &\n\\end{tabular}\nare positive, the double series may be rearranged to make the simple series\n\\[\n\\sum_{xptldfou=0}^{\\infty} 2^{-2 xptldfou}\n\\]\nwhich sums to \\( 4 / 3 \\). The sum of the original double series is therefore \\( 4 / 3 \\). To formalize this argument one must prove that\n\\[\n(qzxwvtnp, hjgrksla) \\mapsto(hjgrksla+qzxwvtnp)^{2}+3 hjgrksla+qzxwvtnp\n\\]\nis a bijection from the set of ordered pairs of non-negative integers to the set of non-negative even integers. This is straightforward but long.\n\nA direct analytic argument can be given as follows: Sum the double series along the diagonals defined by \\( qzxwvtnp+hjgrksla=vrbqmsze \\), and then sum on \\( vrbqmsze \\).\n\nWe find\n\\[\n\\begin{aligned}\n\\sum_{qzxwvtnp=0}^{\\infty} \\sum_{hjgrksla=0}^{\\infty} 2^{-3 hjgrksla-qzxwvtnp-(qzxwvtnp+hjgrksla)^{2}} & =\\sum_{vrbqmsze=0}^{\\infty} \\sum_{hjgrksla=0}^{vrbqmsze} 2^{-2 hjgrksla-vrbqmsze-vrbqmsze^{2}} \\\\\n& =\\sum_{vrbqmsze=0}^{\\infty}\\left[\\frac{4}{3}\\left(1-2^{-2 vrbqmsze-2}\\right) 2^{-vrbqmsze-vrbqmsze^{2}}\\right] \\\\\n& =\\frac{4}{3}\\left[\\sum_{vrbqmsze=0}^{\\infty} 2^{-vrbqmsze-vrbqmsze^{2}}-\\sum_{vrbqmsze=0}^{\\infty} 2^{-(vrbqmsze+1)(vrbqmsze+2)}\\right] \\\\\n& =\\frac{4}{3}\\left[\\sum_{vrbqmsze=0}^{\\infty} 2^{-vrbqmsze(vrbqmsze+1)}-\\sum_{xptldfou=1}^{\\infty} 2^{-xptldfou(xptldfou+1)}\\right] \\\\\n& =4 / 3\n\\end{aligned}\n\\]\n\nAt the third step, the separation of the sum into two sums is permissible because the new sums are convergent."
    },
    "kernel_variant": {
      "question": "Evaluate the absolutely convergent triple series  \n\n\\[\n\\boxed{%\n\\displaystyle \nS=\\sum_{i=0}^{\\infty}\\;\\sum_{j=0}^{\\infty}\\;\\sum_{k=0}^{\\infty}\n2^{-\\bigl[(\\,i+j+k\\,)^2+(i+j+k)+2j+4k\\bigr]}}\n\\]\n\nGive the value of \\(S\\) correct to at least twelve significant decimal digits and give a complete, fully–justified derivation of your result.  Every index manipulation, every change of summation order, and every algebraic step must be rigorously explained.",
      "solution": "Step 1 – Absolute convergence  \nThe exponent is a positive–definite quadratic form, so the general term is bounded above by a constant multiple of \\(2^{-\\,(i+j+k)^2}\\); hence the series is dominated by a convergent Gaussian sum and therefore converges absolutely.  This justifies any rearrangement that follows.\n\n--------------------------------------------------------------------\nStep 2 – Collapse the three indices to one\n\nPut  \n\\[\nn=i+j+k \\qquad(n\\in\\mathbb N).\n\\]\nFor a fixed \\(n\\) the variables \\((i,j,k)\\) range over the two–dimensional simplex  \n\n\\[\n\\mathcal S_n=\\{(i,j,k)\\in\\mathbb N^3\\;:\\;i+j+k=n\\}.\n\\]\n\nBecause the exponent already contains the combination \\(n=i+j+k\\), write\n\n\\[\nS=\\sum_{n=0}^{\\infty}2^{-\\bigl[n^{2}+n\\bigr]}\\;\n        \\sum_{(i,j,k)\\in\\mathcal S_n}2^{-(2j+4k)}.\n\\]\n\nThe inner sum depends on \\(j\\) and \\(k\\) only.  If \\(j\\) is fixed, then \\(k\\) runs from \\(0\\) to \\(n-j\\).  Hence\n\n\\[\nT_n:=\\sum_{(i,j,k)\\in\\mathcal S_n}2^{-(2j+4k)}\n     =\\sum_{j=0}^{n}2^{-2j}\\sum_{k=0}^{\\,n-j}2^{-4k}.\n\\]\n\n--------------------------------------------------------------------\nStep 3 – Evaluate \\(T_n\\)\n\nThe \\(k\\)-sum is a finite geometric series:\n\n\\[\n\\sum_{k=0}^{n-j}2^{-4k}=\\frac{1-2^{-4(n-j+1)}}{1-2^{-4}}\n                     =\\frac{16}{15}\\Bigl(1-2^{-4(n-j+1)}\\Bigr).\n\\]\n\nTherefore  \n\n\\[\nT_n=\\frac{16}{15}\\sum_{j=0}^{n}2^{-2j}\n        \\Bigl(1-2^{-4(n-j+1)}\\Bigr)\n    =\\frac{16}{15}\\Bigl(A_n-B_n\\Bigr),\n\\]\n\nwhere  \n\n\\[\nA_n=\\sum_{j=0}^{n}2^{-2j},\n\\qquad\nB_n=\\sum_{j=0}^{n}2^{-2j}\\,2^{-4(n-j+1)}.\n\\]\n\nBoth are again geometric, and a direct computation gives  \n\n\\[\nA_n=\\frac{4}{3}\\Bigl(1-2^{-2n-2}\\Bigr),\\qquad\nB_n=\\frac13\\Bigl(2^{-2n-2}-2^{-4n-4}\\Bigr).\n\\]\n\nConsequently  \n\n\\[\n\\boxed{\\,T_n=\\frac{64}{45}\n             -\\frac{16}{9}\\,2^{-2n-2}\n             +\\frac{16}{45}\\,2^{-4n-4}\\, }.\n\\]\n\n--------------------------------------------------------------------\nStep 4 – Reduce the whole triple sum to three single series\n\nInsert \\(T_n\\) into the expression for \\(S\\):\n\n\\[\nS=\\sum_{n=0}^{\\infty}2^{-n^{2}-n}\\,T_n\n  =\\frac{64}{45}\\sum_{n=0}^{\\infty}2^{-n^{2}-n}\n  -\\frac{16}{9}\\sum_{n=0}^{\\infty}2^{-n^{2}-3n-2}\n  +\\frac{16}{45}\\sum_{n=0}^{\\infty}2^{-n^{2}-5n-4}.\n\\]\n\n--------------------------------------------------------------------\nStep 5 – Index shifts that make the three series comparable  \n\nDefine the classical “quadratic–Gaussian” series  \n\n\\[\n\\Theta:=\\sum_{m=0}^{\\infty}2^{-m(m+1)}.\n\\]\n\nA convenient re-indexing shows\n\n\\[\n\\begin{aligned}\n\\sum_{n=0}^{\\infty}2^{-n^{2}-n}&=\\Theta,\\\\[4pt]\n\\sum_{n=0}^{\\infty}2^{-n^{2}-3n-2}\n   &=\\sum_{m=1}^{\\infty}2^{-m^{2}-m}\n    =\\Theta-1,\\\\[4pt]\n\\sum_{n=0}^{\\infty}2^{-n^{2}-5n-4}\n   &=\\sum_{m=2}^{\\infty}2^{-m^{2}-m+2}\n    =4\\!\\left(\\Theta-1-\\tfrac14\\right)=4\\Theta-5.\n\\end{aligned}\n\\]\n\n--------------------------------------------------------------------\nStep 6 – Combine all contributions\n\nCollecting the coefficients,\n\n\\[\n\\begin{aligned}\nS &=\n\\frac{64}{45}\\Theta-\\frac{16}{9}\\bigl(\\Theta-1\\bigr)\n+\\frac{16}{45}\\bigl(4\\Theta-5\\bigr)  \\\\[6pt]\n&=\\Theta\\!\\left(\\frac{64}{45}-\\frac{80}{45}+\\frac{64}{45}\\right)\n          +\\Bigl(\\frac{16}{9}-\\frac{80}{45}\\Bigr) \\\\[6pt]\n&=\\frac{16}{15}\\,\\Theta.\n\\end{aligned}\n\\]\n\nA remarkable simplification: the complicated triple sum is just a rational multiple of the classical theta-series \\(\\Theta\\).\n\n--------------------------------------------------------------------\nStep 7 – Numerical evaluation (twelve significant digits)\n\n\\[\n\\begin{aligned}\n\\Theta &=1+2^{-2}+2^{-6}+2^{-12}+2^{-20}+2^{-30}+2^{-42}+\\cdots \\\\\n       &\\approx1.265\\,870\\,095\\,230\\,866\\quad(\\text{error}<10^{-16}),\n\\end{aligned}\n\\]\n\nhence  \n\n\\[\n\\boxed{\\;\nS=\\frac{16}{15}\\,\\Theta\n  \\approx1.350\\,261\\,434\\,913\\;}\\qquad\n(\\text{twelve significant digits}).\n\\]\n\n--------------------------------------------------------------------\nStep 8 – Error control  \nThe tail after the term \\(n=6\\) in \\(\\Theta\\) is bounded by  \n\n\\[\n\\sum_{n=7}^{\\infty}2^{-n(n+1)}\n  <\\sum_{n=7}^{\\infty}2^{-56}\\,2^{-(n-7)}\n  <2^{-56}\\frac{1}{1-\\tfrac12}<10^{-16},\n\\]\n\nso our quoted value of \\(S\\) is correct to at least the requested twelve significant digits.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.521845",
        "was_fixed": false,
        "difficulty_analysis": "•  Extra dimension – The original problem was a double sum; the new variant is a triple sum over a two-dimensional simplex for every fixed \\(n\\).  \n•  More elaborate quadratic form – The exponent \\((i+j+k)^2+(i+j+k)+2j+4k\\) contains cross–terms that force the solver to separate the diagonal part from two different directional penalties (coefficients 2 and 4).  \n•  Multi-stage reduction – One must:\n  1. Prove absolute convergence for a three-variable Gaussian-type series,  \n  2. Collapse three indices to one while keeping track of a two-dimensional combinatorial sum,  \n  3. Evaluate that inner double geometric sum in closed form,  \n  4. Recognise and manipulate three distinct quadratic–Gaussian series by clever index shifts to make them comparable,  \n  5. Perform an exact cancellation that is far from obvious a priori.  \n•  Deeper knowledge – Facility with quadratic forms, finite–simplex enumeration, geometric-series manipulations, theta-series, and rigorous error bounds is required.  \n•  Explicit high-precision value – The solver must not only simplify symbolically but also give a controlled-precision numerical answer, demanding careful truncation-error estimates.\n\nHence the enhanced kernel problem is substantially more technical and conceptually richer than both the original and the first kernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Evaluate the absolutely convergent triple series  \n\n\\[\n\\boxed{%\n\\displaystyle \nS=\\sum_{i=0}^{\\infty}\\;\\sum_{j=0}^{\\infty}\\;\\sum_{k=0}^{\\infty}\n2^{-\\bigl[(\\,i+j+k\\,)^2+(i+j+k)+2j+4k\\bigr]}}\n\\]\n\nGive the value of \\(S\\) correct to at least twelve significant decimal digits and give a complete, fully–justified derivation of your result.  Every index manipulation, every change of summation order, and every algebraic step must be rigorously explained.",
      "solution": "Step 1 – Absolute convergence  \nThe exponent is a positive–definite quadratic form, so the general term is bounded above by a constant multiple of \\(2^{-\\,(i+j+k)^2}\\); hence the series is dominated by a convergent Gaussian sum and therefore converges absolutely.  This justifies any rearrangement that follows.\n\n--------------------------------------------------------------------\nStep 2 – Collapse the three indices to one\n\nPut  \n\\[\nn=i+j+k \\qquad(n\\in\\mathbb N).\n\\]\nFor a fixed \\(n\\) the variables \\((i,j,k)\\) range over the two–dimensional simplex  \n\n\\[\n\\mathcal S_n=\\{(i,j,k)\\in\\mathbb N^3\\;:\\;i+j+k=n\\}.\n\\]\n\nBecause the exponent already contains the combination \\(n=i+j+k\\), write\n\n\\[\nS=\\sum_{n=0}^{\\infty}2^{-\\bigl[n^{2}+n\\bigr]}\\;\n        \\sum_{(i,j,k)\\in\\mathcal S_n}2^{-(2j+4k)}.\n\\]\n\nThe inner sum depends on \\(j\\) and \\(k\\) only.  If \\(j\\) is fixed, then \\(k\\) runs from \\(0\\) to \\(n-j\\).  Hence\n\n\\[\nT_n:=\\sum_{(i,j,k)\\in\\mathcal S_n}2^{-(2j+4k)}\n     =\\sum_{j=0}^{n}2^{-2j}\\sum_{k=0}^{\\,n-j}2^{-4k}.\n\\]\n\n--------------------------------------------------------------------\nStep 3 – Evaluate \\(T_n\\)\n\nThe \\(k\\)-sum is a finite geometric series:\n\n\\[\n\\sum_{k=0}^{n-j}2^{-4k}=\\frac{1-2^{-4(n-j+1)}}{1-2^{-4}}\n                     =\\frac{16}{15}\\Bigl(1-2^{-4(n-j+1)}\\Bigr).\n\\]\n\nTherefore  \n\n\\[\nT_n=\\frac{16}{15}\\sum_{j=0}^{n}2^{-2j}\n        \\Bigl(1-2^{-4(n-j+1)}\\Bigr)\n    =\\frac{16}{15}\\Bigl(A_n-B_n\\Bigr),\n\\]\n\nwhere  \n\n\\[\nA_n=\\sum_{j=0}^{n}2^{-2j},\n\\qquad\nB_n=\\sum_{j=0}^{n}2^{-2j}\\,2^{-4(n-j+1)}.\n\\]\n\nBoth are again geometric, and a direct computation gives  \n\n\\[\nA_n=\\frac{4}{3}\\Bigl(1-2^{-2n-2}\\Bigr),\\qquad\nB_n=\\frac13\\Bigl(2^{-2n-2}-2^{-4n-4}\\Bigr).\n\\]\n\nConsequently  \n\n\\[\n\\boxed{\\,T_n=\\frac{64}{45}\n             -\\frac{16}{9}\\,2^{-2n-2}\n             +\\frac{16}{45}\\,2^{-4n-4}\\, }.\n\\]\n\n--------------------------------------------------------------------\nStep 4 – Reduce the whole triple sum to three single series\n\nInsert \\(T_n\\) into the expression for \\(S\\):\n\n\\[\nS=\\sum_{n=0}^{\\infty}2^{-n^{2}-n}\\,T_n\n  =\\frac{64}{45}\\sum_{n=0}^{\\infty}2^{-n^{2}-n}\n  -\\frac{16}{9}\\sum_{n=0}^{\\infty}2^{-n^{2}-3n-2}\n  +\\frac{16}{45}\\sum_{n=0}^{\\infty}2^{-n^{2}-5n-4}.\n\\]\n\n--------------------------------------------------------------------\nStep 5 – Index shifts that make the three series comparable  \n\nDefine the classical “quadratic–Gaussian” series  \n\n\\[\n\\Theta:=\\sum_{m=0}^{\\infty}2^{-m(m+1)}.\n\\]\n\nA convenient re-indexing shows\n\n\\[\n\\begin{aligned}\n\\sum_{n=0}^{\\infty}2^{-n^{2}-n}&=\\Theta,\\\\[4pt]\n\\sum_{n=0}^{\\infty}2^{-n^{2}-3n-2}\n   &=\\sum_{m=1}^{\\infty}2^{-m^{2}-m}\n    =\\Theta-1,\\\\[4pt]\n\\sum_{n=0}^{\\infty}2^{-n^{2}-5n-4}\n   &=\\sum_{m=2}^{\\infty}2^{-m^{2}-m+2}\n    =4\\!\\left(\\Theta-1-\\tfrac14\\right)=4\\Theta-5.\n\\end{aligned}\n\\]\n\n--------------------------------------------------------------------\nStep 6 – Combine all contributions\n\nCollecting the coefficients,\n\n\\[\n\\begin{aligned}\nS &=\n\\frac{64}{45}\\Theta-\\frac{16}{9}\\bigl(\\Theta-1\\bigr)\n+\\frac{16}{45}\\bigl(4\\Theta-5\\bigr)  \\\\[6pt]\n&=\\Theta\\!\\left(\\frac{64}{45}-\\frac{80}{45}+\\frac{64}{45}\\right)\n          +\\Bigl(\\frac{16}{9}-\\frac{80}{45}\\Bigr) \\\\[6pt]\n&=\\frac{16}{15}\\,\\Theta.\n\\end{aligned}\n\\]\n\nA remarkable simplification: the complicated triple sum is just a rational multiple of the classical theta-series \\(\\Theta\\).\n\n--------------------------------------------------------------------\nStep 7 – Numerical evaluation (twelve significant digits)\n\n\\[\n\\begin{aligned}\n\\Theta &=1+2^{-2}+2^{-6}+2^{-12}+2^{-20}+2^{-30}+2^{-42}+\\cdots \\\\\n       &\\approx1.265\\,870\\,095\\,230\\,866\\quad(\\text{error}<10^{-16}),\n\\end{aligned}\n\\]\n\nhence  \n\n\\[\n\\boxed{\\;\nS=\\frac{16}{15}\\,\\Theta\n  \\approx1.350\\,261\\,434\\,913\\;}\\qquad\n(\\text{twelve significant digits}).\n\\]\n\n--------------------------------------------------------------------\nStep 8 – Error control  \nThe tail after the term \\(n=6\\) in \\(\\Theta\\) is bounded by  \n\n\\[\n\\sum_{n=7}^{\\infty}2^{-n(n+1)}\n  <\\sum_{n=7}^{\\infty}2^{-56}\\,2^{-(n-7)}\n  <2^{-56}\\frac{1}{1-\\tfrac12}<10^{-16},\n\\]\n\nso our quoted value of \\(S\\) is correct to at least the requested twelve significant digits.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.436832",
        "was_fixed": false,
        "difficulty_analysis": "•  Extra dimension – The original problem was a double sum; the new variant is a triple sum over a two-dimensional simplex for every fixed \\(n\\).  \n•  More elaborate quadratic form – The exponent \\((i+j+k)^2+(i+j+k)+2j+4k\\) contains cross–terms that force the solver to separate the diagonal part from two different directional penalties (coefficients 2 and 4).  \n•  Multi-stage reduction – One must:\n  1. Prove absolute convergence for a three-variable Gaussian-type series,  \n  2. Collapse three indices to one while keeping track of a two-dimensional combinatorial sum,  \n  3. Evaluate that inner double geometric sum in closed form,  \n  4. Recognise and manipulate three distinct quadratic–Gaussian series by clever index shifts to make them comparable,  \n  5. Perform an exact cancellation that is far from obvious a priori.  \n•  Deeper knowledge – Facility with quadratic forms, finite–simplex enumeration, geometric-series manipulations, theta-series, and rigorous error bounds is required.  \n•  Explicit high-precision value – The solver must not only simplify symbolically but also give a controlled-precision numerical answer, demanding careful truncation-error estimates.\n\nHence the enhanced kernel problem is substantially more technical and conceptually richer than both the original and the first kernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "calculation",
  "iteratively_fixed": true
}