{
  "index": "1942-A-3",
  "type": "ANA",
  "tag": [
    "ANA",
    "ALG"
  ],
  "difficulty": "",
  "question": "\\begin{array}{l}\n\\text { 3. Is the following series convergent or divergent? }\\\\\n1+\\frac{1}{2} \\cdot \\frac{19}{7}+\\frac{2!}{3^{2}}\\left(\\frac{19}{7}\\right)^{2}+\\frac{3!}{4^{3}}\\left(\\frac{19}{7}\\right)^{3}+\\frac{4!}{5^{4}}\\left(\\frac{19}{7}\\right)^{4}+\\cdots\n\\end{array}",
  "solution": "Solution. Use the ratio test. Let\n\\[\na_{n}=\\frac{(n-1)!}{n^{n-1}}\\left(\\frac{19}{7}\\right)^{n-1}, \\quad a_{n+1}=\\frac{n!}{(n+1)^{n}}\\left(\\frac{19}{7}\\right)^{n} .\n\\]\n\nThen\n\\[\nR_{n}=\\frac{a_{n+1}}{a_{n}}=\\frac{n^{n}}{(n+1)^{n}} \\frac{19}{7}=\\frac{1}{\\left(1+\\frac{1}{n}\\right)^{n}} \\frac{19}{7},\n\\]\nand\n\\[\n\\lim _{n \\rightarrow \\infty} R_{n}=\\frac{19}{7} \\lim \\frac{1}{\\left(1+\\frac{1}{n}\\right)^{n}}=\\frac{19}{7} \\frac{1}{e} .\n\\]\n\nSince \\( 19 / 7<2.715 \\) and \\( e>2.718,19 / 7 e<1 \\) and the series converges.",
  "vars": [
    "a_n",
    "a_n+1",
    "R_n",
    "n"
  ],
  "params": [],
  "sci_consts": [
    "e"
  ],
  "variants": {
    "descriptive_long": {
      "map": {
        "a_n": "termseq",
        "a_n+1": "termnext",
        "R_n": "ratioseq",
        "n": "indexvar"
      },
      "question": "\\begin{array}{l}\n\\text { 3. Is the following series convergent or divergent? }\\\\\n1+\\frac{1}{2} \\cdot \\frac{19}{7}+\\frac{2!}{3^{2}}\\left(\\frac{19}{7}\\right)^{2}+\\frac{3!}{4^{3}}\\left(\\frac{19}{7}\\right)^{3}+\\frac{4!}{5^{4}}\\left(\\frac{19}{7}\\right)^{4}+\\cdots\n\\end{array}",
      "solution": "Solution. Use the ratio test. Let\n\\[\ntermseq=\\frac{(indexvar-1)!}{indexvar^{indexvar-1}}\\left(\\frac{19}{7}\\right)^{indexvar-1}, \\quad termnext=\\frac{indexvar!}{(indexvar+1)^{indexvar}}\\left(\\frac{19}{7}\\right)^{indexvar} .\n\\]\n\nThen\n\\[\nratioseq=\\frac{termnext}{termseq}=\\frac{indexvar^{indexvar}}{(indexvar+1)^{indexvar}} \\frac{19}{7}=\\frac{1}{\\left(1+\\frac{1}{indexvar}\\right)^{indexvar}} \\frac{19}{7},\n\\]\nand\n\\[\n\\lim _{indexvar \\rightarrow \\infty} ratioseq=\\frac{19}{7} \\lim \\frac{1}{\\left(1+\\frac{1}{indexvar}\\right)^{indexvar}}=\\frac{19}{7} \\frac{1}{e} .\n\\]\n\nSince \\( 19 / 7<2.715 \\) and \\( e>2.718,19 / 7 e<1 \\) and the series converges."
    },
    "descriptive_long_confusing": {
      "map": {
        "a_n": "shoreline",
        "a_n+1": "crosswind",
        "R_n": "drumstick",
        "n": "pineapple"
      },
      "question": "\\begin{array}{l}\n\\text { 3. Is the following series convergent or divergent? }\\\\\n1+\\frac{1}{2} \\cdot \\frac{19}{7}+\\frac{2!}{3^{2}}\\left(\\frac{19}{7}\\right)^{2}+\\frac{3!}{4^{3}}\\left(\\frac{19}{7}\\right)^{3}+\\frac{4!}{5^{4}}\\left(\\frac{19}{7}\\right)^{4}+\\cdots\n\\end{array}",
      "solution": "Solution. Use the ratio test. Let\n\\[\nshoreline=\\frac{(pineapple-1)!}{pineapple^{pineapple-1}}\\left(\\frac{19}{7}\\right)^{pineapple-1}, \\quad crosswind=\\frac{pineapple!}{(pineapple+1)^{pineapple}}\\left(\\frac{19}{7}\\right)^{pineapple} .\n\\]\n\nThen\n\\[\ndrumstick=\\frac{crosswind}{shoreline}=\\frac{pineapple^{pineapple}}{(pineapple+1)^{pineapple}} \\frac{19}{7}=\\frac{1}{\\left(1+\\frac{1}{pineapple}\\right)^{pineapple}} \\frac{19}{7},\n\\]\nand\n\\[\n\\lim _{pineapple \\rightarrow \\infty} drumstick=\\frac{19}{7} \\lim \\frac{1}{\\left(1+\\frac{1}{pineapple}\\right)^{pineapple}}=\\frac{19}{7} \\frac{1}{e} .\n\\]\n\nSince \\( 19 / 7<2.715 \\) and \\( e>2.718,19 / 7 e<1 \\) and the series converges."
    },
    "descriptive_long_misleading": {
      "map": {
        "a_n": "constant",
        "a_n+1": "steadyval",
        "R_n": "difference",
        "n": "continuous"
      },
      "question": "\\begin{array}{l}\n\\text { 3. Is the following series convergent or divergent? }\\\\\n1+\\frac{1}{2} \\cdot \\frac{19}{7}+\\frac{2!}{3^{2}}\\left(\\frac{19}{7}\\right)^{2}+\\frac{3!}{4^{3}}\\left(\\frac{19}{7}\\right)^{3}+\\frac{4!}{5^{4}}\\left(\\frac{19}{7}\\right)^{4}+\\cdots\n\\end{array}",
      "solution": "Solution. Use the ratio test. Let\n\\[\nconstant=\\frac{(continuous-1)!}{continuous^{continuous-1}}\\left(\\frac{19}{7}\\right)^{continuous-1}, \\quad steadyval=\\frac{continuous!}{(continuous+1)^{continuous}}\\left(\\frac{19}{7}\\right)^{continuous} .\n\\]\n\nThen\n\\[\ndifference=\\frac{steadyval}{constant}=\\frac{continuous^{continuous}}{(continuous+1)^{continuous}} \\frac{19}{7}=\\frac{1}{\\left(1+\\frac{1}{continuous}\\right)^{continuous}} \\frac{19}{7},\n\\]\nand\n\\[\n\\lim _{continuous \\rightarrow \\infty} difference=\\frac{19}{7} \\lim \\frac{1}{\\left(1+\\frac{1}{continuous}\\right)^{continuous}}=\\frac{19}{7} \\frac{1}{e} .\n\\]\n\nSince \\( 19 / 7<2.715 \\) and \\( e>2.718,19 / 7 e<1 \\) and the series converges."
    },
    "garbled_string": {
      "map": {
        "a_n": "qzxwvtnp",
        "a_n+1": "hjgrksla",
        "R_n": "vekdfbrj",
        "n": "bnxesrty"
      },
      "question": "\\begin{array}{l}\n\\text { 3. Is the following series convergent or divergent? }\\\\\n1+\\frac{1}{2} \\cdot \\frac{19}{7}+\\frac{2!}{3^{2}}\\left(\\frac{19}{7}\\right)^{2}+\\frac{3!}{4^{3}}\\left(\\frac{19}{7}\\right)^{3}+\\frac{4!}{5^{4}}\\left(\\frac{19}{7}\\right)^{4}+\\cdots\n\\end{array}",
      "solution": "Solution. Use the ratio test. Let\n\\[\nqzxwvtnp=\\frac{(bnxesrty-1)!}{bnxesrty^{bnxesrty-1}}\\left(\\frac{19}{7}\\right)^{bnxesrty-1}, \\quad hjgrksla=\\frac{bnxesrty!}{(bnxesrty+1)^{bnxesrty}}\\left(\\frac{19}{7}\\right)^{bnxesrty} .\n\\]\n\nThen\n\\[\nvekdfbrj=\\frac{hjgrksla}{qzxwvtnp}=\\frac{bnxesrty^{bnxesrty}}{(bnxesrty+1)^{bnxesrty}} \\frac{19}{7}=\\frac{1}{\\left(1+\\frac{1}{bnxesrty}\\right)^{bnxesrty}} \\frac{19}{7},\n\\]\nand\n\\[\n\\lim _{bnxesrty \\rightarrow \\infty} vekdfbrj=\\frac{19}{7} \\lim \\frac{1}{\\left(1+\\frac{1}{bnxesrty}\\right)^{bnxesrty}}=\\frac{19}{7} \\frac{1}{e} .\n\\]\n\nSince \\( 19 / 7<2.715 \\) and \\( e>2.718,19 / 7 e<1 \\) and the series converges."
    },
    "kernel_variant": {
      "question": "Let three real parameters satisfy  \n\\[\n\\alpha>0,\\qquad \\beta\\ge 0,\\qquad \\lambda>0 .\n\\]\n\nDefine the alternating series  \n\\[\nS(\\alpha,\\beta,\\lambda)\\;=\\;\n\\sum_{n=1}^{\\infty}\n(-1)^{\\,n-1}\\;\n\\dfrac{\\bigl[(n-1)!\\bigr]^{\\alpha}\\,\n\\bigl(\\tfrac{11}{3}\\bigr)^{\\lambda(n-1)}}{n^{\\,n+\\beta}} .\n\\]\n\nPartition the parameter space \\((\\alpha,\\beta,\\lambda)\\) into the three **mutually exclusive** regions  \n\nA. absolute convergence of the series,  \n\nB. conditional (but not absolute) convergence,  \n\nC. divergence.  \n\nFor each region give explicit inequalities in \\(\\alpha,\\beta,\\lambda\\) and supply a complete rigorous proof – purely analytic, with no numerical experimentation or computer algebra.\n\n────────────────────────────────────────────────────────────────────────",
      "solution": "Throughout we set  \n\\[\nc:=\\Bigl(\\tfrac{11}{3}\\Bigr)^{\\lambda}\\;>\\;1,\\qquad \nr:=\\frac{c}{e},\\qquad \n\\lambda_{c}:=\\frac{1}{\\ln(11/3)}\\;( \\text{so that } r=1 \\Longleftrightarrow \\lambda=\\lambda_{c}).\n\\]\n\nDenote  \n\\[\na_n=(-1)^{\\,n-1}\\frac{[(n-1)!]^{\\alpha}\\,c^{\\,n-1}}{n^{\\,n+\\beta}},\\qquad n\\ge 1.\n\\]\n\nStep 1 – Root test away from the plane \\(\\alpha=1\\).\n\nUsing Stirling’s approximation  \n\\[\n(n-1)!=(n-1)^{\\,n-1}e^{-(n-1)}\\sqrt{2\\pi(n-1)}\\bigl(1+o(1)\\bigr)\\quad(n\\to\\infty),\n\\]\nwe obtain\n\\[\n\\lvert a_n\\rvert^{1/n}\n=\\Bigl(\\tfrac{11}{3}\\Bigr)^{\\lambda}e^{-\\alpha}\\,n^{\\alpha-1}\\bigl(1+o(1)\\bigr)\\qquad(n\\to\\infty).\n\\]\nHence  \n\\[\n\\lim_{n\\to\\infty}\\lvert a_n\\rvert^{1/n}\n=\n\\begin{cases}\n0, & 0<\\alpha<1,\\\\[4pt]\nr, & \\alpha=1,\\\\[4pt]\n\\infty, & \\alpha>1.\n\\end{cases}\n\\]\n\nConsequences of the root test:\n\n(i) \\(0<\\alpha<1\\): the limit is \\(0\\Rightarrow\\) **absolute convergence** for every \\(\\beta,\\lambda\\).\n\n(ii) \\(\\alpha>1\\): the limit is \\(\\infty\\Rightarrow\\) **divergence** for every \\(\\beta,\\lambda\\).\n\n(iii) \\(\\alpha=1\\): the limit equals \\(r\\).  If \\(r<1\\) (\\(\\lambda<\\lambda_{c}\\)) we again have absolute convergence; if \\(r>1\\) (\\(\\lambda>\\lambda_{c}\\)) we have divergence.  \nExactly when \\(r=1\\) (\\(\\lambda=\\lambda_{c}\\)) the root test is inconclusive and a finer analysis is required.\n\nThe remaining work is therefore restricted to the line  \n\\[\n\\alpha=1,\\qquad \\lambda=\\lambda_{c}.\n\\]\n\nStep 2 – Precise asymptotics on \\(\\alpha=1,\\lambda=\\lambda_{c}\\).\n\nPut \\(\\alpha=1\\) and \\(\\lambda=\\lambda_{c}\\,( \\Rightarrow c=e)\\).  Then\n\\[\na_n=(-1)^{\\,n-1}\\frac{(n-1)!}{n^{\\,n+\\beta}}e^{\\,n-1}.\n\\]\nInsert Stirling’s formula:\n\n\\[\n\\lvert a_n\\rvert\n=\\sqrt{2\\pi(n-1)}\\left(\\frac{n-1}{e}\\right)^{n-1}\n\\frac{e^{\\,n-1}}{n^{\\,n+\\beta}}\n=\\sqrt{2\\pi(n-1)}\\left(1-\\frac1n\\right)^{n-1}n^{-1-\\beta}.\n\\]\n\nSince \\(\\left(1-\\frac1n\\right)^{n-1}=e^{-1}\\bigl(1+O(\\tfrac1n)\\bigr)\\) and  \n\\(\\sqrt{2\\pi(n-1)}=\\sqrt{2\\pi}\\,n^{1/2}\\bigl(1+O(\\tfrac1n)\\bigr)\\),\n\\[\n\\boxed{\\;\n\\lvert a_n\\rvert\n=\nC\\,n^{-\\beta-\\tfrac12}\\bigl(1+o(1)\\bigr),\\qquad \nC:=\\frac{\\sqrt{2\\pi}}{e}.\n\\;}\n\\tag{1}\n\\]\n\nStep 3 – Absolute convergence on the critical line.\n\nBecause the general term is asymptotically equivalent to \\(C\\,n^{-\\beta-\\tfrac12}\\),\n\n• if \\(\\beta+\\tfrac12>1\\) (that is \\(\\beta> \\tfrac12\\)) the comparison test with the \\(p\\)-series \\(\\sum n^{-p}\\) (\\(p=\\beta+\\tfrac12>1\\)) gives **absolute convergence**;\n\n• if \\(\\beta+\\tfrac12\\le 1\\) (\\(0\\le\\beta\\le\\tfrac12\\)) the series of absolute values diverges.\n\nStep 4 – Conditional convergence for \\(0\\le\\beta\\le\\tfrac12\\).\n\nFor \\(0\\le\\beta\\le\\tfrac12\\) we still have \\(\\lvert a_n\\rvert\\to 0\\) and, by (1),\n\\[\n\\frac{\\lvert a_{n+1}\\rvert}{\\lvert a_n\\rvert}\n=\\frac{(n+1)^{-\\beta-\\tfrac12}(1+o(1))}{n^{-\\beta-\\tfrac12}(1+o(1))}\n=\\left(\\frac{n}{n+1}\\right)^{\\beta+\\tfrac12}(1+o(1))<1\\quad(n\\gg1),\n\\]\nso \\(\\lvert a_n\\rvert\\) is eventually decreasing.  \nThe Leibniz criterion therefore yields convergence of the alternating series, while absolute convergence fails; hence the series is **conditionally convergent**.\n\nStep 5 – Final classification.\n\nA.  Absolute convergence  \n\\[\n\\begin{cases}\n0<\\alpha<1, & \\beta\\ge 0,\\;\\lambda>0,\\\\[4pt]\n\\alpha=1,\\;\\lambda<\\lambda_{c}, & \\beta\\ge 0,\\\\[4pt]\n\\alpha=1,\\;\\lambda=\\lambda_{c}, & \\beta>\\tfrac12.\n\\end{cases}\n\\]\n\nB.  Conditional (but not absolute) convergence  \n\\[\n\\alpha=1,\\qquad \\lambda=\\lambda_{c},\\qquad 0\\le\\beta\\le\\tfrac12.\n\\]\n\nC.  Divergence  \n\\[\n\\begin{cases}\n\\alpha>1, & \\beta\\ge 0,\\;\\lambda>0,\\\\[4pt]\n\\alpha=1,\\;\\lambda>\\lambda_{c}, & \\beta\\ge 0.\n\\end{cases}\n\\]\n\nThe three regions are mutually exclusive and cover the entire parameter space \\(\\{\\alpha>0,\\;\\beta\\ge0,\\;\\lambda>0\\}\\), so the trichotomy is complete.\n\n────────────────────────────────────────────────────────────────────────",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.392162",
        "was_fixed": false,
        "difficulty_analysis": "1. Multi-parameter space.  Unlike the single-series originals, the enhanced problem requires a full three–parameter classification, forcing competitors to keep track of interacting inequalities rather than producing a single yes/no answer.\n\n2. Higher-order asymptotics.  The boundary case α=1 necessitates a second-order use of Stirling’s formula and delicate limiting arguments beyond a routine ratio test.\n\n3. Alternating versus absolute behaviour.  Competitors must invoke both the root-test and the alternating-series test and know exactly when each applies, as well as recognise the subtle cancellation that changes e^{−n} into 1/n^{1+β} on the critical line λ=λ_c.\n\n4. Layered logical structure.  Properly organising cases (α<1, α=1, α>1; then sub-cases in α=1) and proving exclusivity and completeness is substantially more intricate than the original ratio-test exercise.\n\n5. Parameter threshold in closed form.  Extracting λ_c = 1/ln(11/3) demands algebraic manipulation of inequalities arising from the asymptotics, adding yet another conceptual step.\n\nAll these features jointly make the enhanced variant significantly more technically demanding than both the original problem and the current kernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Let three real parameters satisfy  \n\\[\n\\alpha>0,\\qquad \\beta\\ge 0,\\qquad \\lambda>0 .\n\\]\n\nDefine the alternating series  \n\\[\nS(\\alpha,\\beta,\\lambda)\\;=\\;\n\\sum_{n=1}^{\\infty}\n(-1)^{\\,n-1}\\;\n\\dfrac{\\bigl[(n-1)!\\bigr]^{\\alpha}\\,\n\\bigl(\\tfrac{11}{3}\\bigr)^{\\lambda(n-1)}}{n^{\\,n+\\beta}} .\n\\]\n\nPartition the parameter space \\((\\alpha,\\beta,\\lambda)\\) into the three **mutually exclusive** regions  \n\nA. absolute convergence of the series,  \n\nB. conditional (but not absolute) convergence,  \n\nC. divergence.  \n\nFor each region give explicit inequalities in \\(\\alpha,\\beta,\\lambda\\) and supply a complete rigorous proof – purely analytic, with no numerical experimentation or computer algebra.\n\n────────────────────────────────────────────────────────────────────────",
      "solution": "Throughout we set  \n\\[\nc:=\\Bigl(\\tfrac{11}{3}\\Bigr)^{\\lambda}\\;>\\;1,\\qquad \nr:=\\frac{c}{e},\\qquad \n\\lambda_{c}:=\\frac{1}{\\ln(11/3)}\\;( \\text{so that } r=1 \\Longleftrightarrow \\lambda=\\lambda_{c}).\n\\]\n\nDenote  \n\\[\na_n=(-1)^{\\,n-1}\\frac{[(n-1)!]^{\\alpha}\\,c^{\\,n-1}}{n^{\\,n+\\beta}},\\qquad n\\ge 1.\n\\]\n\nStep 1 – Root test away from the plane \\(\\alpha=1\\).\n\nUsing Stirling’s approximation  \n\\[\n(n-1)!=(n-1)^{\\,n-1}e^{-(n-1)}\\sqrt{2\\pi(n-1)}\\bigl(1+o(1)\\bigr)\\quad(n\\to\\infty),\n\\]\nwe obtain\n\\[\n\\lvert a_n\\rvert^{1/n}\n=\\Bigl(\\tfrac{11}{3}\\Bigr)^{\\lambda}e^{-\\alpha}\\,n^{\\alpha-1}\\bigl(1+o(1)\\bigr)\\qquad(n\\to\\infty).\n\\]\nHence  \n\\[\n\\lim_{n\\to\\infty}\\lvert a_n\\rvert^{1/n}\n=\n\\begin{cases}\n0, & 0<\\alpha<1,\\\\[4pt]\nr, & \\alpha=1,\\\\[4pt]\n\\infty, & \\alpha>1.\n\\end{cases}\n\\]\n\nConsequences of the root test:\n\n(i) \\(0<\\alpha<1\\): the limit is \\(0\\Rightarrow\\) **absolute convergence** for every \\(\\beta,\\lambda\\).\n\n(ii) \\(\\alpha>1\\): the limit is \\(\\infty\\Rightarrow\\) **divergence** for every \\(\\beta,\\lambda\\).\n\n(iii) \\(\\alpha=1\\): the limit equals \\(r\\).  If \\(r<1\\) (\\(\\lambda<\\lambda_{c}\\)) we again have absolute convergence; if \\(r>1\\) (\\(\\lambda>\\lambda_{c}\\)) we have divergence.  \nExactly when \\(r=1\\) (\\(\\lambda=\\lambda_{c}\\)) the root test is inconclusive and a finer analysis is required.\n\nThe remaining work is therefore restricted to the line  \n\\[\n\\alpha=1,\\qquad \\lambda=\\lambda_{c}.\n\\]\n\nStep 2 – Precise asymptotics on \\(\\alpha=1,\\lambda=\\lambda_{c}\\).\n\nPut \\(\\alpha=1\\) and \\(\\lambda=\\lambda_{c}\\,( \\Rightarrow c=e)\\).  Then\n\\[\na_n=(-1)^{\\,n-1}\\frac{(n-1)!}{n^{\\,n+\\beta}}e^{\\,n-1}.\n\\]\nInsert Stirling’s formula:\n\n\\[\n\\lvert a_n\\rvert\n=\\sqrt{2\\pi(n-1)}\\left(\\frac{n-1}{e}\\right)^{n-1}\n\\frac{e^{\\,n-1}}{n^{\\,n+\\beta}}\n=\\sqrt{2\\pi(n-1)}\\left(1-\\frac1n\\right)^{n-1}n^{-1-\\beta}.\n\\]\n\nSince \\(\\left(1-\\frac1n\\right)^{n-1}=e^{-1}\\bigl(1+O(\\tfrac1n)\\bigr)\\) and  \n\\(\\sqrt{2\\pi(n-1)}=\\sqrt{2\\pi}\\,n^{1/2}\\bigl(1+O(\\tfrac1n)\\bigr)\\),\n\\[\n\\boxed{\\;\n\\lvert a_n\\rvert\n=\nC\\,n^{-\\beta-\\tfrac12}\\bigl(1+o(1)\\bigr),\\qquad \nC:=\\frac{\\sqrt{2\\pi}}{e}.\n\\;}\n\\tag{1}\n\\]\n\nStep 3 – Absolute convergence on the critical line.\n\nBecause the general term is asymptotically equivalent to \\(C\\,n^{-\\beta-\\tfrac12}\\),\n\n• if \\(\\beta+\\tfrac12>1\\) (that is \\(\\beta> \\tfrac12\\)) the comparison test with the \\(p\\)-series \\(\\sum n^{-p}\\) (\\(p=\\beta+\\tfrac12>1\\)) gives **absolute convergence**;\n\n• if \\(\\beta+\\tfrac12\\le 1\\) (\\(0\\le\\beta\\le\\tfrac12\\)) the series of absolute values diverges.\n\nStep 4 – Conditional convergence for \\(0\\le\\beta\\le\\tfrac12\\).\n\nFor \\(0\\le\\beta\\le\\tfrac12\\) we still have \\(\\lvert a_n\\rvert\\to 0\\) and, by (1),\n\\[\n\\frac{\\lvert a_{n+1}\\rvert}{\\lvert a_n\\rvert}\n=\\frac{(n+1)^{-\\beta-\\tfrac12}(1+o(1))}{n^{-\\beta-\\tfrac12}(1+o(1))}\n=\\left(\\frac{n}{n+1}\\right)^{\\beta+\\tfrac12}(1+o(1))<1\\quad(n\\gg1),\n\\]\nso \\(\\lvert a_n\\rvert\\) is eventually decreasing.  \nThe Leibniz criterion therefore yields convergence of the alternating series, while absolute convergence fails; hence the series is **conditionally convergent**.\n\nStep 5 – Final classification.\n\nA.  Absolute convergence  \n\\[\n\\begin{cases}\n0<\\alpha<1, & \\beta\\ge 0,\\;\\lambda>0,\\\\[4pt]\n\\alpha=1,\\;\\lambda<\\lambda_{c}, & \\beta\\ge 0,\\\\[4pt]\n\\alpha=1,\\;\\lambda=\\lambda_{c}, & \\beta>\\tfrac12.\n\\end{cases}\n\\]\n\nB.  Conditional (but not absolute) convergence  \n\\[\n\\alpha=1,\\qquad \\lambda=\\lambda_{c},\\qquad 0\\le\\beta\\le\\tfrac12.\n\\]\n\nC.  Divergence  \n\\[\n\\begin{cases}\n\\alpha>1, & \\beta\\ge 0,\\;\\lambda>0,\\\\[4pt]\n\\alpha=1,\\;\\lambda>\\lambda_{c}, & \\beta\\ge 0.\n\\end{cases}\n\\]\n\nThe three regions are mutually exclusive and cover the entire parameter space \\(\\{\\alpha>0,\\;\\beta\\ge0,\\;\\lambda>0\\}\\), so the trichotomy is complete.\n\n────────────────────────────────────────────────────────────────────────",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.337261",
        "was_fixed": false,
        "difficulty_analysis": "1. Multi-parameter space.  Unlike the single-series originals, the enhanced problem requires a full three–parameter classification, forcing competitors to keep track of interacting inequalities rather than producing a single yes/no answer.\n\n2. Higher-order asymptotics.  The boundary case α=1 necessitates a second-order use of Stirling’s formula and delicate limiting arguments beyond a routine ratio test.\n\n3. Alternating versus absolute behaviour.  Competitors must invoke both the root-test and the alternating-series test and know exactly when each applies, as well as recognise the subtle cancellation that changes e^{−n} into 1/n^{1+β} on the critical line λ=λ_c.\n\n4. Layered logical structure.  Properly organising cases (α<1, α=1, α>1; then sub-cases in α=1) and proving exclusivity and completeness is substantially more intricate than the original ratio-test exercise.\n\n5. Parameter threshold in closed form.  Extracting λ_c = 1/ln(11/3) demands algebraic manipulation of inequalities arising from the asymptotics, adding yet another conceptual step.\n\nAll these features jointly make the enhanced variant significantly more technically demanding than both the original problem and the current kernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}