{
  "index": "1963-A-4",
  "type": "ANA",
  "tag": [
    "ANA",
    "ALG",
    "NT"
  ],
  "difficulty": "",
  "question": "4. Let \\( \\left\\{a_{n}\\right\\} \\) be a sequence of positive real numbers. Show that\n\\[\n\\lim _{n \\rightarrow \\infty} \\sup n\\left(\\frac{1+a_{n+1}}{a_{n}}-1\\right) \\geq 1\n\\]\n\nShow that the number 1 on the right-hand side of this inequality cannot be replaced by any larger number. (The symbol lim sup is sometimes written lim.)",
  "solution": "Solution. Suppose that for some fixed integer \\( k \\),\n\\[\nn\\left(\\frac{1+a_{n+1}}{a_{n}}-1\\right) \\leq 1\n\\]\nfor all \\( n \\geq k \\). Then\n\\[\n\\begin{aligned}\n1+a_{n+1} & \\leq \\frac{n+1}{n} a_{n}, \\\\\n\\frac{a_{n}}{n} & \\geq \\frac{1}{n+1}+\\frac{a_{n+1}}{n+1}\n\\end{aligned}\n\\]\nfor \\( n \\geq k \\). Accordingly we have\n\\[\n\\begin{aligned}\n\\frac{a_{k}}{k} & \\geq \\frac{1}{k+1}+\\frac{a_{k+1}}{k+1} \\geq \\frac{1}{k+1}+\\frac{1}{k+2}+\\frac{a_{k+2}}{k+2} \\\\\n& \\geq \\frac{1}{k+1}+\\frac{1}{k+2}+\\cdots+\\frac{1}{k+p}+\\frac{a_{k+p}}{k+p} \\\\\n& \\geq \\frac{1}{k+1}+\\frac{1}{k+2}+\\cdots+\\frac{1}{k+p}\n\\end{aligned}\n\\]\nfor each \\( p \\). However, this is impossible since the harmonic series diverges.\nThus for any \\( k \\) there exists an \\( n \\geq k \\) such that\n\\[\nn\\left(\\frac{1+a_{n+1}}{a_{n}}-1\\right)>1 .\n\\]\n\nTherefore\n\\[\n\\limsup _{n \\rightarrow \\infty} n\\left(\\frac{1+a_{n+1}}{a_{n}}-1\\right) \\geq 1 .\n\\]\n\nIf we take \\( a_{n}=n \\log n \\), then\n\\[\n\\begin{aligned}\nn\\left(\\frac{1+a_{n+1}}{a_{n}}-1\\right) & =\\frac{1+(n+1) \\log (n+1)-n \\log n}{\\log n} \\\\\n& =\\frac{1}{\\log n}\\left[1+n \\log \\frac{n+1}{n}+\\log (n+1)\\right] \\\\\n& \\leq \\frac{1}{\\log n}[2+\\log (n+1)]\n\\end{aligned}\n\\]\nsince \\( \\log (1+x) \\leq x \\) for all \\( x>-1 \\). Since the right side has limit 1 , we have, for the particular sequence,\n\\[\n\\limsup _{n \\rightarrow \\infty} n\\left(\\frac{1+a_{n+1}}{a_{n}}-1\\right) \\leq 1 .\n\\]\n\nThus we cannot increase the bound 1.\nWe could also take \\( a_{n}=n^{1+\\epsilon} \\) where \\( \\epsilon>0 \\) and obtain\n\\[\n\\limsup _{n-\\infty} n\\left(\\frac{1+a_{n+1}}{a_{n}}-1\\right)=1+\\epsilon .\n\\]",
  "vars": [
    "n",
    "a_n",
    "a_n+1",
    "a_k",
    "a_k+1",
    "a_k+2",
    "a_k+p",
    "k",
    "p",
    "x"
  ],
  "params": [
    "\\\\epsilon"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "n": "indexvar",
        "a_n": "seqterm",
        "a_n+1": "seqnext",
        "a_k": "seqtermk",
        "a_k+1": "seqnextk",
        "a_k+2": "seqnextk2",
        "a_k+p": "seqkplusp",
        "k": "startidx",
        "p": "iteratep",
        "x": "varxplaceholder",
        "\\epsilon": "smalltol"
      },
      "question": "4. Let \\( \\{seqterm\\} \\) be a sequence of positive real numbers. Show that\n\\[\n\\lim _{indexvar \\rightarrow \\infty} \\sup indexvar\\left(\\frac{1+seqnext}{seqterm}-1\\right) \\geq 1\n\\]\n\nShow that the number 1 on the right-hand side of this inequality cannot be replaced by any larger number. (The symbol lim sup is sometimes written lim.)",
      "solution": "Solution. Suppose that for some fixed integer \\( startidx \\),\n\\[\nindexvar\\left(\\frac{1+seqnext}{seqterm}-1\\right) \\leq 1\n\\]\nfor all \\( indexvar \\geq startidx \\). Then\n\\[\n\\begin{aligned}\n1+seqnext & \\leq \\frac{indexvar+1}{indexvar} \\, seqterm, \\\\\n\\frac{seqterm}{indexvar} & \\geq \\frac{1}{indexvar+1}+\\frac{seqnext}{indexvar+1}\n\\end{aligned}\n\\]\nfor \\( indexvar \\geq startidx \\). Accordingly we have\n\\[\n\\begin{aligned}\n\\frac{seqtermk}{startidx} & \\geq \\frac{1}{startidx+1}+\\frac{seqnextk}{startidx+1} \\geq \\frac{1}{startidx+1}+\\frac{1}{startidx+2}+\\frac{seqnextk2}{startidx+2} \\\\\n& \\geq \\frac{1}{startidx+1}+\\frac{1}{startidx+2}+\\cdots+\\frac{1}{startidx+iteratep}+\\frac{seqkplusp}{startidx+iteratep} \\\\\n& \\geq \\frac{1}{startidx+1}+\\frac{1}{startidx+2}+\\cdots+\\frac{1}{startidx+iteratep}\n\\end{aligned}\n\\]\nfor each \\( iteratep \\). However, this is impossible since the harmonic series diverges.\nThus for any \\( startidx \\) there exists an \\( indexvar \\geq startidx \\) such that\n\\[\nindexvar\\left(\\frac{1+seqnext}{seqterm}-1\\right)>1 .\n\\]\n\nTherefore\n\\[\n\\limsup _{indexvar \\rightarrow \\infty} indexvar\\left(\\frac{1+seqnext}{seqterm}-1\\right) \\geq 1 .\n\\]\n\nIf we take \\( seqterm=indexvar \\log indexvar \\), then\n\\[\n\\begin{aligned}\nindexvar\\left(\\frac{1+seqnext}{seqterm}-1\\right) & =\\frac{1+(indexvar+1) \\log (indexvar+1)-indexvar \\log indexvar}{\\log indexvar} \\\\\n& =\\frac{1}{\\log indexvar}\\left[1+indexvar \\log \\frac{indexvar+1}{indexvar}+\\log (indexvar+1)\\right] \\\\\n& \\leq \\frac{1}{\\log indexvar}[2+\\log (indexvar+1)]\n\\end{aligned}\n\\]\nsince \\( \\log (1+varxplaceholder) \\leq varxplaceholder \\) for all \\( varxplaceholder>-1 \\). Since the right side has limit 1, we have, for the particular sequence,\n\\[\n\\limsup _{indexvar \\rightarrow \\infty} indexvar\\left(\\frac{1+seqnext}{seqterm}-1\\right) \\leq 1 .\n\\]\n\nThus we cannot increase the bound 1.\nWe could also take \\( seqterm=indexvar^{1+smalltol} \\) where \\( smalltol>0 \\) and obtain\n\\[\n\\limsup _{indexvar-\\infty} indexvar\\left(\\frac{1+seqnext}{seqterm}-1\\right)=1+smalltol .\n\\]"
    },
    "descriptive_long_confusing": {
      "map": {
        "n": "jackalope",
        "a_n": "driftwood",
        "a_n+1": "moonlight",
        "a_k": "sandcastle",
        "a_k+1": "peppermint",
        "a_k+2": "horseshoe",
        "a_k+p": "raincloud",
        "k": "tumbleweed",
        "p": "goldfish",
        "x": "jellybean",
        "\\epsilon": "buttercup"
      },
      "question": "4. Let \\( \\left\\{driftwood\\right\\} \\) be a sequence of positive real numbers. Show that\n\\[\n\\lim _{jackalope \\rightarrow \\infty} \\sup jackalope\\left(\\frac{1+moonlight}{driftwood}-1\\right) \\geq 1\n\\]\n\nShow that the number 1 on the right-hand side of this inequality cannot be replaced by any larger number. (The symbol lim sup is sometimes written lim.)",
      "solution": "Solution. Suppose that for some fixed integer \\( tumbleweed \\),\n\\[\njackalope\\left(\\frac{1+moonlight}{driftwood}-1\\right) \\leq 1\n\\]\nfor all \\( jackalope \\geq tumbleweed \\). Then\n\\[\n\\begin{aligned}\n1+moonlight & \\leq \\frac{jackalope+1}{jackalope} driftwood, \\\\\n\\frac{driftwood}{jackalope} & \\geq \\frac{1}{jackalope+1}+\\frac{moonlight}{jackalope+1}\n\\end{aligned}\n\\]\nfor \\( jackalope \\geq tumbleweed \\). Accordingly we have\n\\[\n\\begin{aligned}\n\\frac{sandcastle}{tumbleweed} & \\geq \\frac{1}{tumbleweed+1}+\\frac{peppermint}{tumbleweed+1} \\geq \\frac{1}{tumbleweed+1}+\\frac{1}{tumbleweed+2}+\\frac{horseshoe}{tumbleweed+2} \\\\\n& \\geq \\frac{1}{tumbleweed+1}+\\frac{1}{tumbleweed+2}+\\cdots+\\frac{1}{tumbleweed+goldfish}+\\frac{raincloud}{tumbleweed+goldfish} \\\\\n& \\geq \\frac{1}{tumbleweed+1}+\\frac{1}{tumbleweed+2}+\\cdots+\\frac{1}{tumbleweed+goldfish}\n\\end{aligned}\n\\]\nfor each \\( goldfish \\). However, this is impossible since the harmonic series diverges.\nThus for any \\( tumbleweed \\) there exists an \\( jackalope \\geq tumbleweed \\) such that\n\\[\njackalope\\left(\\frac{1+moonlight}{driftwood}-1\\right)>1 .\n\\]\n\nTherefore\n\\[\n\\limsup _{jackalope \\rightarrow \\infty} jackalope\\left(\\frac{1+moonlight}{driftwood}-1\\right) \\geq 1 .\n\\]\n\nIf we take \\( driftwood=jackalope \\log jackalope \\), then\n\\[\n\\begin{aligned}\njackalope\\left(\\frac{1+moonlight}{driftwood}-1\\right) & =\\frac{1+(jackalope+1) \\log (jackalope+1)-jackalope \\log jackalope}{\\log jackalope} \\\\\n& =\\frac{1}{\\log jackalope}\\left[1+jackalope \\log \\frac{jackalope+1}{jackalope}+\\log (jackalope+1)\\right] \\\\\n& \\leq \\frac{1}{\\log jackalope}[2+\\log (jackalope+1)]\n\\end{aligned}\n\\]\nsince \\( \\log (1+jellybean) \\leq jellybean \\) for all \\( jellybean>-1 \\). Since the right side has limit 1 , we have, for the particular sequence,\n\\[\n\\limsup _{jackalope \\rightarrow \\infty} jackalope\\left(\\frac{1+moonlight}{driftwood}-1\\right) \\leq 1 .\n\\]\n\nThus we cannot increase the bound 1.\nWe could also take \\( driftwood=jackalope^{1+buttercup} \\) where \\( buttercup>0 \\) and obtain\n\\[\n\\limsup _{jackalope-\\infty} jackalope\\left(\\frac{1+moonlight}{driftwood}-1\\right)=1+buttercup .\n\\]"
    },
    "descriptive_long_misleading": {
      "map": {
        "n": "continuumindex",
        "a_n": "fixedvalue",
        "a_n+1": "previousvalue",
        "a_k": "stabledata",
        "a_k+1": "stagnantdatum",
        "a_k+2": "motionlessinfo",
        "a_k+p": "unchangingrecord",
        "k": "variableindex",
        "p": "stableoffset",
        "x": "constantinput",
        "\\\\epsilon": "largenumber"
      },
      "question": "Let \\( \\left\\{fixedvalue\\right\\} \\) be a sequence of positive real numbers. Show that\n\\[\n\\lim _{continuumindex \\rightarrow \\infty} \\sup continuumindex\\left(\\frac{1+previousvalue}{fixedvalue}-1\\right) \\geq 1\n\\]\n\nShow that the number 1 on the right-hand side of this inequality cannot be replaced by any larger number.",
      "solution": "Solution. Suppose that for some fixed integer \\( variableindex \\),\n\\[\ncontinuumindex\\left(\\frac{1+previousvalue}{fixedvalue}-1\\right) \\leq 1\n\\]\nfor all \\( continuumindex \\geq variableindex \\). Then\n\\[\n\\begin{aligned}\n1+previousvalue & \\leq \\frac{continuumindex+1}{continuumindex} fixedvalue, \\\\\n\\frac{fixedvalue}{continuumindex} & \\geq \\frac{1}{continuumindex+1}+\\frac{previousvalue}{continuumindex+1}\n\\end{aligned}\n\\]\nfor \\( continuumindex \\geq variableindex \\). Accordingly we have\n\\[\n\\begin{aligned}\n\\frac{stabledata}{variableindex} & \\geq \\frac{1}{variableindex+1}+\\frac{stagnantdatum}{variableindex+1} \\geq \\frac{1}{variableindex+1}+\\frac{1}{variableindex+2}+\\frac{motionlessinfo}{variableindex+2} \\\\\n& \\geq \\frac{1}{variableindex+1}+\\frac{1}{variableindex+2}+\\cdots+\\frac{1}{variableindex+stableoffset}+\\frac{unchangingrecord}{variableindex+stableoffset} \\\\\n& \\geq \\frac{1}{variableindex+1}+\\frac{1}{variableindex+2}+\\cdots+\\frac{1}{variableindex+stableoffset}\n\\end{aligned}\n\\]\nfor each \\( stableoffset \\). However, this is impossible since the harmonic series diverges.\nThus for any \\( variableindex \\) there exists an \\( continuumindex \\geq variableindex \\) such that\n\\[\ncontinuumindex\\left(\\frac{1+previousvalue}{fixedvalue}-1\\right)>1 .\n\\]\n\nTherefore\n\\[\n\\limsup _{continuumindex \\rightarrow \\infty} continuumindex\\left(\\frac{1+previousvalue}{fixedvalue}-1\\right) \\geq 1 .\n\\]\n\nIf we take \\( fixedvalue=continuumindex \\log continuumindex \\), then\n\\[\n\\begin{aligned}\ncontinuumindex\\left(\\frac{1+previousvalue}{fixedvalue}-1\\right) & =\\frac{1+(continuumindex+1) \\log (continuumindex+1)-continuumindex \\log continuumindex}{\\log continuumindex} \\\\\n& =\\frac{1}{\\log continuumindex}\\left[1+continuumindex \\log \\frac{continuumindex+1}{continuumindex}+\\log (continuumindex+1)\\right] \\\\\n& \\leq \\frac{1}{\\log continuumindex}[2+\\log (continuumindex+1)]\n\\end{aligned}\n\\]\nsince \\( \\log (1+constantinput) \\leq constantinput \\) for all \\( constantinput>-1 \\). Since the right side has limit 1 , we have, for the particular sequence,\n\\[\n\\limsup _{continuumindex \\rightarrow \\infty} continuumindex\\left(\\frac{1+previousvalue}{fixedvalue}-1\\right) \\leq 1 .\n\\]\n\nThus we cannot increase the bound 1.\nWe could also take \\( fixedvalue=continuumindex^{1+largenumber} \\) where \\( largenumber>0 \\) and obtain\n\\[\n\\limsup _{continuumindex-\\infty} continuumindex\\left(\\frac{1+previousvalue}{fixedvalue}-1\\right)=1+largenumber .\n\\]"
    },
    "garbled_string": {
      "map": {
        "n": "zlmqrtyh",
        "a_n": "vxcnsdpe",
        "a_n+1": "kljhrwtu",
        "a_k": "mbqrzdyo",
        "a_k+1": "wnfzpxis",
        "a_k+2": "ydrgvkqe",
        "a_k+p": "gqsldvma",
        "k": "psnhxdaw",
        "p": "tlkquorz",
        "x": "rnfvqzie",
        "\\epsilon": "cjpwexhm"
      },
      "question": "4. Let \\( \\left\\{vxcnsdpe\\right\\} \\) be a sequence of positive real numbers. Show that\n\\[\n\\lim _{zlmqrtyh \\rightarrow \\infty} \\sup zlmqrtyh\\left(\\frac{1+kljhrwtu}{vxcnsdpe}-1\\right) \\geq 1\n\\]\n\nShow that the number 1 on the right-hand side of this inequality cannot be replaced by any larger number. (The symbol lim sup is sometimes written lim.)",
      "solution": "Solution. Suppose that for some fixed integer \\( psnhxdaw \\),\n\\[\nzlmqrtyh\\left(\\frac{1+kljhrwtu}{vxcnsdpe}-1\\right) \\leq 1\n\\]\nfor all \\( zlmqrtyh \\geq psnhxdaw \\). Then\n\\[\n\\begin{aligned}\n1+kljhrwtu & \\leq \\frac{zlmqrtyh+1}{zlmqrtyh} vxcnsdpe, \\\\\n\\frac{vxcnsdpe}{zlmqrtyh} & \\geq \\frac{1}{zlmqrtyh+1}+\\frac{kljhrwtu}{zlmqrtyh+1}\n\\end{aligned}\n\\]\nfor \\( zlmqrtyh \\geq psnhxdaw \\). Accordingly we have\n\\[\n\\begin{aligned}\n\\frac{mbqrzdyo}{psnhxdaw} & \\geq \\frac{1}{psnhxdaw+1}+\\frac{wnfzpxis}{psnhxdaw+1} \\geq \\frac{1}{psnhxdaw+1}+\\frac{1}{psnhxdaw+2}+\\frac{ydrgvkqe}{psnhxdaw+2} \\\\\n& \\geq \\frac{1}{psnhxdaw+1}+\\frac{1}{psnhxdaw+2}+\\cdots+\\frac{1}{psnhxdaw+tlkquorz}+\\frac{gqsldvma}{psnhxdaw+tlkquorz} \\\\\n& \\geq \\frac{1}{psnhxdaw+1}+\\frac{1}{psnhxdaw+2}+\\cdots+\\frac{1}{psnhxdaw+tlkquorz}\n\\end{aligned}\n\\]\nfor each \\( tlkquorz \\). However, this is impossible since the harmonic series diverges.\nThus for any \\( psnhxdaw \\) there exists an \\( zlmqrtyh \\geq psnhxdaw \\) such that\n\\[\nzlmqrtyh\\left(\\frac{1+kljhrwtu}{vxcnsdpe}-1\\right)>1 .\n\\]\n\nTherefore\n\\[\n\\limsup _{zlmqrtyh \\rightarrow \\infty} zlmqrtyh\\left(\\frac{1+kljhrwtu}{vxcnsdpe}-1\\right) \\geq 1 .\n\\]\n\nIf we take \\( vxcnsdpe=zlmqrtyh \\log zlmqrtyh \\), then\n\\[\n\\begin{aligned}\nzlmqrtyh\\left(\\frac{1+kljhrwtu}{vxcnsdpe}-1\\right) & =\\frac{1+(zlmqrtyh+1) \\log (zlmqrtyh+1)-zlmqrtyh \\log zlmqrtyh}{\\log zlmqrtyh} \\\\\n& =\\frac{1}{\\log zlmqrtyh}\\left[1+zlmqrtyh \\log \\frac{zlmqrtyh+1}{zlmqrtyh}+\\log (zlmqrtyh+1)\\right] \\\\\n& \\leq \\frac{1}{\\log zlmqrtyh}[2+\\log (zlmqrtyh+1)]\n\\end{aligned}\n\\]\nsince \\( \\log (1+rnfvqzie) \\leq rnfvqzie \\) for all \\( rnfvqzie>-1 \\). Since the right side has limit 1, we have, for the particular sequence,\n\\[\n\\limsup _{zlmqrtyh \\rightarrow \\infty} zlmqrtyh\\left(\\frac{1+kljhrwtu}{vxcnsdpe}-1\\right) \\leq 1 .\n\\]\n\nThus we cannot increase the bound 1.\nWe could also take \\( vxcnsdpe=zlmqrtyh^{1+cjpwexhm} \\) where \\( cjpwexhm>0 \\) and obtain\n\\[\n\\limsup _{zlmqrtyh-\\infty} zlmqrtyh\\left(\\frac{1+kljhrwtu}{vxcnsdpe}-1\\right)=1+cjpwexhm .\n\\]"
    },
    "kernel_variant": {
      "question": "Let $m\\ge 1$ be a fixed positive integer and let $\\,(a_{n})_{n\\ge 1}$ be an arbitrary sequence of positive real numbers.  \nFor every $n\\ge 1$ put  \n\n\\[\nE_{n}\\;:=\\;n\\Bigl(\\tfrac{m+a_{n+m}}{a_{n}}-1\\Bigr).\\tag{$\\star$}\n\\]\n\n1. (Universal lower bound)  Prove that  \n\n\\[\n\\limsup_{n\\to\\infty} E_{n}\\;\\ge\\;m. \\tag{1}\n\\]\n\n2. (Sharpness of the constant)  Show that the number $m$ in (1) is best possible: for every $\\varepsilon>0$ there exists a positive sequence $(a_{n})$ such that  \n\n\\[\n\\limsup_{n\\to\\infty} E_{n}\\;\\le\\;m+\\varepsilon .\n\\]\n\n(In particular one can have $\\limsup_{n\\to\\infty}E_{n}=m$.)\n\n3. (Asymptotic structure of the extremal sequences)  \nWrite $L_{n}:=a_{n}/n$ $(n\\ge 1)$.  Prove that  \n\n\\[\n\\lim_{n\\to\\infty}E_{n}=m \\tag{2}\n\\]\n\nholds if and only if  \n\n\\[\n\\text{(a)}\\;L_{n}\\longrightarrow\\infty ,\n\\qquad\n\\text{(b)}\\;(n+m)\\bigl(L_{n+m}/L_{n}-1\\bigr)\\longrightarrow 0. \\tag{3}\n\\]\n\nHence the sequences for which (2) is true are precisely those of the form  \n\n\\[\na_{n}=nL_{n}\\quad\\text{with}\\quad L_{n}\\longrightarrow\\infty\n\\;\\text{ and }\\;L_{n+m}/L_{n}\\longrightarrow 1 .\n\\]\n\nGive a few explicit examples (for instance $a_{n}=n(\\log n)^{\\alpha}$ with $\\alpha>0$, or  \n$a_{n}=n\\exp[(\\log\\log n)^{\\beta}]$ with $0<\\beta<1$) to illustrate the variety of such extremal sequences.",
      "solution": "Throughout we set  \n\n\\[\nb_{n}:=\\frac{a_{n}}{n}\\qquad(n\\ge 1),\n\\quad\\text{so that}\\quad a_{n}=nb_{n}>0 .\n\\]\n\nA direct substitution in $(\\star)$ gives the key identity  \n\n\\[\nE_{n}=m+\\frac{m}{b_{n}}+(n+m)\\Bigl(\\frac{b_{n+m}}{b_{n}}-1\\Bigr)\\qquad(n\\ge 1). \\tag{4}\n\\]\n\n────────────────────────────────────────  \nPart 1.  Proof of the universal bound (1)  \n────────────────────────────────────────  \n\nAssume, on the contrary, that $\\limsup_{n\\to\\infty}E_{n}=m-2\\delta$ for some $\\delta>0$.  \nThen there is $N$ such that  \n\n\\[\nE_{n}\\le m-\\delta\\qquad(n\\ge N). \\tag{5}\n\\]\n\nFrom (4) we obtain, for $n\\ge N$,  \n\n\\[\n\\frac{m}{b_{n}}+(n+m)\\Bigl(\\frac{b_{n+m}}{b_{n}}-1\\Bigr)\\le-\\delta. \\tag{6}\n\\]\n\nBecause the first summand in (6) is non–negative, the second one is negative, hence $b_{n+m}<b_{n}$.  \nRearranging (6) gives  \n\n\\[\nb_{n}-b_{n+m}\\;\\ge\\;\\frac{m}{n+m}. \\tag{7}\n\\]\n\nLet $n_{k}:=N+km$ $(k\\ge 0)$.  \nSumming (7) for $k=0,1,\\dots,p-1$ yields  \n\n\\[\nb_{n_{p}}\\;\\le\\;b_{N}-m\\sum_{k=1}^{p}\\frac{1}{n_{k}}\\;.\n\\]\n\nThe harmonic series diverges, so the right–hand side eventually becomes negative, contradicting $b_{n}>0$.  \nHence (5) is impossible and (1) is established. ∎\n\n\n\n────────────────────────────────────────  \nPart 2.  Sharpness of the constant  \n────────────────────────────────────────  \n\n(a) Example with $\\displaystyle\\limsup E_{n}=m$.  \nLet $a_{n}=n(\\log n)^{\\alpha}$ with $\\alpha>0$ and $n\\ge 3$.  \nBy the Taylor expansion $\\log(n+m)=\\log n+\\dfrac{m}{n}+O(1/n^{2})$ we get  \n\n\\[\n\\frac{a_{n+m}}{a_{n}}\n      =1+\\frac{m}{n}+\\frac{\\alpha m}{n\\log n}+O\\!\\left(\\frac{1}{n^{2}}\\right),\n\\qquad\n\\frac{m}{a_{n}}=O\\!\\left(\\frac{1}{n(\\log n)^{\\alpha}}\\right).\n\\]\n\nSubstituting in (4) gives  \n\n\\[\nE_{n}=m+\\frac{m}{(\\log n)^{\\alpha}}+\\frac{\\alpha m}{\\log n}+O\\!\\left(\\frac{1}{n}\\right)=m+o(1),\n\\]\nhence $\\limsup E_{n}=m$.\n\n(b) Example with $\\displaystyle\\limsup E_{n}=m+\\varepsilon$.  \nFix $\\varepsilon>0$ and take $a_{n}=n^{1+\\varepsilon/m}$.  Then  \n\n\\[\n\\frac{a_{n+m}}{a_{n}}\n      =(1+\\tfrac{m}{n})^{1+\\varepsilon/m}\n      =1+\\Bigl(1+\\tfrac{\\varepsilon}{m}\\Bigr)\\frac{m}{n}+O\\!\\left(\\frac{1}{n^{2}}\\right),\n\\qquad\n\\frac{m}{a_{n}}=O\\!\\bigl(n^{-1-\\varepsilon/m}\\bigr),\n\\]\nand (4) yields  \n\n\\[\nE_{n}=m+\\varepsilon+O\\!\\bigl(n^{-\\varepsilon/m}\\bigr)\\longrightarrow m+\\varepsilon .\n\\]\n\nBecause $\\varepsilon>0$ is arbitrary, the constant $m$ in (1) cannot be increased. ∎\n\n\n\n────────────────────────────────────────  \nPart 3.  Asymptotic description of the extremals  \n────────────────────────────────────────  \n\nWe prove that (2) is equivalent to the pair of conditions (3).\n\nNecessity.  \nAssume $E_{n}\\to m$.  From (4) we have  \n\n\\[\nE_{n}-m=\\frac{m}{b_{n}}+(n+m)\\Bigl(\\frac{b_{n+m}}{b_{n}}-1\\Bigr)\\xrightarrow[n\\to\\infty]{}0. \\tag{8}\n\\]\n\nStep 1.  Show that $b_{n}\\to\\infty$ (i.e. (3a)).  \nSuppose, to the contrary, that $b_{n}$ is not unbounded.  \nThen there exist $B>0$ and an infinite subsequence $(n_{k})$ with $b_{n_{k}}\\le B$.  \nLet  \n\n\\[\n\\varepsilon:=\\frac{m}{2B}>0 .\n\\]\n\nBecause $E_{n}\\to m$, we can choose this subsequence so that  \n\n\\[\n\\lvert E_{n_{k}}-m\\rvert<\\varepsilon \\qquad\\text{for all }k. \\tag{9}\n\\]\n\nApplying (8) to these indices and using $b_{n_{k}}\\le B$ gives  \n\n\\[\n(n_{k}+m)\\Bigl(1-\\tfrac{b_{n_{k}+m}}{b_{n_{k}}}\\Bigr)\n      =\\frac{m}{b_{n_{k}}}-(E_{n_{k}}-m)\n      \\ge\\frac{m}{B}-\\varepsilon\n      =\\varepsilon . \\tag{10}\n\\]\n\nHence  \n\n\\[\nb_{n_{k}+m}\\;\\le\\;b_{n_{k}}\\Bigl(1-\\frac{\\varepsilon}{n_{k}+m}\\Bigr). \\tag{11}\n\\]\n\nIterating (11) over the arithmetic progression $n_{k,r}:=n_{k}+rm$ $(r\\ge 0)$ gives  \n\n\\[\nb_{n_{k,r}}\\;\\le\\;B\n\\exp\\!\\Bigl(-\\varepsilon\\sum_{j=0}^{r-1}\\frac{1}{n_{k}+jm+m}\\Bigr)\n\\longrightarrow 0\\quad(r\\to\\infty). \\tag{12}\n\\]\n\nChoose $r$ so large that  \n\n\\[\nb_{n_{k,r}}\\;<\\;\\frac{m}{2(n_{k,r}+m)}. \\tag{13}\n\\]\n\nReturning to (4) with $n=n_{k,r}$ and using $\\dfrac{b_{n+m}}{b_{n}}>0$ we obtain  \n\n\\[\nE_{n_{k,r}}-m\n      \\;\\ge\\;\\frac{m}{b_{n_{k,r}}}-(n_{k,r}+m)\n      \\;>\\;2(n_{k,r}+m)-(n_{k,r}+m)\n      \\;=\\;n_{k,r}+m .\n\\]\n\nThus $E_{n_{k,r}}-m$ is arbitrarily large, contradicting $E_{n}\\to m$.  \nTherefore the assumption was false and $b_{n}\\to\\infty$.  This proves (3a).\n\nStep 2.  Deduce (3b).  \nSince $b_{n}\\to\\infty$, the first term on the right–hand side of (8) tends to $0$; hence  \n\n\\[\n(n+m)\\Bigl(\\tfrac{b_{n+m}}{b_{n}}-1\\Bigr)\\longrightarrow 0 ,\n\\]\nwhich is exactly (3b).  Necessity is established.\n\nSufficiency.  \nConversely, if (3a)–(3b) hold, then $\\dfrac{m}{b_{n}}\\to 0$ and the last term in (4) also tends to $0$, so $E_{n}\\to m$. ∎\n\n\n\n────────────────────────────────────────  \nExamples of extremal sequences  \n────────────────────────────────────────  \n\n1. $a_{n}=n(\\log n)^{\\alpha}$, $\\alpha>0$.  \n   Then $L_{n}=(\\log n)^{\\alpha}\\to\\infty$, and a direct calculation gives  \n\\[\n(n+m)\\bigl(L_{n+m}/L_{n}-1\\bigr)\n      =\\frac{\\alpha m}{\\log n}+O\\!\\bigl(\\tfrac{1}{(\\log n)^{2}}\\bigr)\\longrightarrow 0.\n\\]\n\n2. $a_{n}=n\\exp\\!\\bigl[(\\log\\log n)^{\\beta}\\bigr]$, $0<\\beta<1$.  \n   Here $\\log L_{n}=(\\log\\log n)^{\\beta}$, and  \n\\[\n\\log\\!\\frac{L_{n+m}}{L_{n}}\n      =(\\log\\log(n+m))^{\\beta}-(\\log\\log n)^{\\beta}\n      =O\\!\\Bigl(\\frac{1}{n(\\log n)^{1-\\beta}}\\Bigr),\n\\]\nso $(3)$ is satisfied.\n\n3. More generally, any $a_{n}=nL_{n}$ with $L_{n}$ slowly varying (with step $m$) and $L_{n}\\to\\infty$ fulfils $E_{n}\\to m$, and every sequence with this property is of that form. ∎",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.542888",
        "was_fixed": false,
        "difficulty_analysis": "• Higher-order interaction:   \n  The expression now links the term aₙ to a_{n+m} across an arbitrary positive\n  offset m.  The proof must control the evolution of the sequence in jumps of\n  length m, rather than in consecutive terms.\n\n• Divergent series with stride m:  \n  Deriving the contradiction requires building an m-step harmonic series\n  ∑_{j}1/(k+jm), not the ordinary harmonic series that sufficed in the\n  original problem.  Handling this necessitates a careful index-shifting\n  argument and a non-trivial comparison with the usual ∑1/ℓ.\n\n• Parameter dependence:  \n  Both the inequality and the optimal constant depend explicitly on m.  One\n  must show simultaneously that  \n  (a) the lower bound scales exactly with m, and  \n  (b) the bound is tight for every m, by constructing appropriate families of\n  test sequences.  This introduces an additional layer of quantification and\n  verification absent from the original kernel.\n\n• More intricate recursion:  \n  Instead of a single forward estimate, the proof demands an m-step recursive\n  inequality and its iteration, leading to a telescoping sum with variable\n  denominators.  Careful bookkeeping of these denominators is essential to\n  guarantee divergence.\n\nOverall, the enhanced variant forces competitors to generalise the harmonic-series\nargument, manage non-unit step sizes, and track a parameter that permeates every\nstage of the reasoning, making the problem substantially more technical and\nconceptually demanding than the original."
      }
    },
    "original_kernel_variant": {
      "question": "Let $m\\ge 1$ be a fixed positive integer and let $\\,(a_{n})_{n\\ge 1}$ be an arbitrary sequence of positive real numbers.  \nFor every $n\\ge 1$ put  \n\n\\[\nE_{n}\\;:=\\;n\\Bigl(\\tfrac{m+a_{n+m}}{a_{n}}-1\\Bigr).\\tag{$\\star$}\n\\]\n\n1. (Universal lower bound)  Prove that  \n\n\\[\n\\limsup_{n\\to\\infty} E_{n}\\;\\ge\\;m. \\tag{1}\n\\]\n\n2. (Sharpness of the constant)  Show that the number $m$ in (1) is best possible: for every $\\varepsilon>0$ there exists a positive sequence $(a_{n})$ such that  \n\n\\[\n\\limsup_{n\\to\\infty} E_{n}\\;\\le\\;m+\\varepsilon .\n\\]\n\n(In particular one can have $\\limsup_{n\\to\\infty}E_{n}=m$.)\n\n3. (Asymptotic structure of the extremal sequences)  \nWrite $L_{n}:=a_{n}/n$ $(n\\ge 1)$.  Prove that  \n\n\\[\n\\lim_{n\\to\\infty}E_{n}=m \\tag{2}\n\\]\n\nholds if and only if  \n\n\\[\n\\text{(a)}\\;L_{n}\\longrightarrow\\infty ,\n\\qquad\n\\text{(b)}\\;(n+m)\\bigl(L_{n+m}/L_{n}-1\\bigr)\\longrightarrow 0. \\tag{3}\n\\]\n\nHence the sequences for which (2) is true are precisely those of the form  \n\n\\[\na_{n}=nL_{n}\\quad\\text{with}\\quad L_{n}\\longrightarrow\\infty\n\\;\\text{ and }\\;L_{n+m}/L_{n}\\longrightarrow 1 .\n\\]\n\nGive a few explicit examples (for instance $a_{n}=n(\\log n)^{\\alpha}$ with $\\alpha>0$, or  \n$a_{n}=n\\exp[(\\log\\log n)^{\\beta}]$ with $0<\\beta<1$) to illustrate the variety of such extremal sequences.",
      "solution": "Throughout we set  \n\n\\[\nb_{n}:=\\frac{a_{n}}{n}\\qquad(n\\ge 1),\n\\quad\\text{so that}\\quad a_{n}=nb_{n}>0 .\n\\]\n\nA direct substitution in $(\\star)$ gives the key identity  \n\n\\[\nE_{n}=m+\\frac{m}{b_{n}}+(n+m)\\Bigl(\\frac{b_{n+m}}{b_{n}}-1\\Bigr)\\qquad(n\\ge 1). \\tag{4}\n\\]\n\n────────────────────────────────────────  \nPart 1.  Proof of the universal bound (1)  \n────────────────────────────────────────  \n\nAssume, on the contrary, that $\\limsup_{n\\to\\infty}E_{n}=m-2\\delta$ for some $\\delta>0$.  \nThen there is $N$ such that  \n\n\\[\nE_{n}\\le m-\\delta\\qquad(n\\ge N). \\tag{5}\n\\]\n\nFrom (4) we obtain, for $n\\ge N$,  \n\n\\[\n\\frac{m}{b_{n}}+(n+m)\\Bigl(\\frac{b_{n+m}}{b_{n}}-1\\Bigr)\\le-\\delta. \\tag{6}\n\\]\n\nBecause the first summand in (6) is non–negative, the second one is negative, hence $b_{n+m}<b_{n}$.  \nRearranging (6) gives  \n\n\\[\nb_{n}-b_{n+m}\\;\\ge\\;\\frac{m}{n+m}. \\tag{7}\n\\]\n\nLet $n_{k}:=N+km$ $(k\\ge 0)$.  \nSumming (7) for $k=0,1,\\dots,p-1$ yields  \n\n\\[\nb_{n_{p}}\\;\\le\\;b_{N}-m\\sum_{k=1}^{p}\\frac{1}{n_{k}}\\;.\n\\]\n\nThe harmonic series diverges, so the right–hand side eventually becomes negative, contradicting $b_{n}>0$.  \nHence (5) is impossible and (1) is established. ∎\n\n\n\n────────────────────────────────────────  \nPart 2.  Sharpness of the constant  \n────────────────────────────────────────  \n\n(a) Example with $\\displaystyle\\limsup E_{n}=m$.  \nLet $a_{n}=n(\\log n)^{\\alpha}$ with $\\alpha>0$ and $n\\ge 3$.  \nBy the Taylor expansion $\\log(n+m)=\\log n+\\dfrac{m}{n}+O(1/n^{2})$ we get  \n\n\\[\n\\frac{a_{n+m}}{a_{n}}\n      =1+\\frac{m}{n}+\\frac{\\alpha m}{n\\log n}+O\\!\\left(\\frac{1}{n^{2}}\\right),\n\\qquad\n\\frac{m}{a_{n}}=O\\!\\left(\\frac{1}{n(\\log n)^{\\alpha}}\\right).\n\\]\n\nSubstituting in (4) gives  \n\n\\[\nE_{n}=m+\\frac{m}{(\\log n)^{\\alpha}}+\\frac{\\alpha m}{\\log n}+O\\!\\left(\\frac{1}{n}\\right)=m+o(1),\n\\]\nhence $\\limsup E_{n}=m$.\n\n(b) Example with $\\displaystyle\\limsup E_{n}=m+\\varepsilon$.  \nFix $\\varepsilon>0$ and take $a_{n}=n^{1+\\varepsilon/m}$.  Then  \n\n\\[\n\\frac{a_{n+m}}{a_{n}}\n      =(1+\\tfrac{m}{n})^{1+\\varepsilon/m}\n      =1+\\Bigl(1+\\tfrac{\\varepsilon}{m}\\Bigr)\\frac{m}{n}+O\\!\\left(\\frac{1}{n^{2}}\\right),\n\\qquad\n\\frac{m}{a_{n}}=O\\!\\bigl(n^{-1-\\varepsilon/m}\\bigr),\n\\]\nand (4) yields  \n\n\\[\nE_{n}=m+\\varepsilon+O\\!\\bigl(n^{-\\varepsilon/m}\\bigr)\\longrightarrow m+\\varepsilon .\n\\]\n\nBecause $\\varepsilon>0$ is arbitrary, the constant $m$ in (1) cannot be increased. ∎\n\n\n\n────────────────────────────────────────  \nPart 3.  Asymptotic description of the extremals  \n────────────────────────────────────────  \n\nWe prove that (2) is equivalent to the pair of conditions (3).\n\nNecessity.  \nAssume $E_{n}\\to m$.  From (4) we have  \n\n\\[\nE_{n}-m=\\frac{m}{b_{n}}+(n+m)\\Bigl(\\frac{b_{n+m}}{b_{n}}-1\\Bigr)\\xrightarrow[n\\to\\infty]{}0. \\tag{8}\n\\]\n\nStep 1.  Show that $b_{n}\\to\\infty$ (i.e. (3a)).  \nSuppose, to the contrary, that $b_{n}$ is not unbounded.  \nThen there exist $B>0$ and an infinite subsequence $(n_{k})$ with $b_{n_{k}}\\le B$.  \nLet  \n\n\\[\n\\varepsilon:=\\frac{m}{2B}>0 .\n\\]\n\nBecause $E_{n}\\to m$, we can choose this subsequence so that  \n\n\\[\n\\lvert E_{n_{k}}-m\\rvert<\\varepsilon \\qquad\\text{for all }k. \\tag{9}\n\\]\n\nApplying (8) to these indices and using $b_{n_{k}}\\le B$ gives  \n\n\\[\n(n_{k}+m)\\Bigl(1-\\tfrac{b_{n_{k}+m}}{b_{n_{k}}}\\Bigr)\n      =\\frac{m}{b_{n_{k}}}-(E_{n_{k}}-m)\n      \\ge\\frac{m}{B}-\\varepsilon\n      =\\varepsilon . \\tag{10}\n\\]\n\nHence  \n\n\\[\nb_{n_{k}+m}\\;\\le\\;b_{n_{k}}\\Bigl(1-\\frac{\\varepsilon}{n_{k}+m}\\Bigr). \\tag{11}\n\\]\n\nIterating (11) over the arithmetic progression $n_{k,r}:=n_{k}+rm$ $(r\\ge 0)$ gives  \n\n\\[\nb_{n_{k,r}}\\;\\le\\;B\n\\exp\\!\\Bigl(-\\varepsilon\\sum_{j=0}^{r-1}\\frac{1}{n_{k}+jm+m}\\Bigr)\n\\longrightarrow 0\\quad(r\\to\\infty). \\tag{12}\n\\]\n\nChoose $r$ so large that  \n\n\\[\nb_{n_{k,r}}\\;<\\;\\frac{m}{2(n_{k,r}+m)}. \\tag{13}\n\\]\n\nReturning to (4) with $n=n_{k,r}$ and using $\\dfrac{b_{n+m}}{b_{n}}>0$ we obtain  \n\n\\[\nE_{n_{k,r}}-m\n      \\;\\ge\\;\\frac{m}{b_{n_{k,r}}}-(n_{k,r}+m)\n      \\;>\\;2(n_{k,r}+m)-(n_{k,r}+m)\n      \\;=\\;n_{k,r}+m .\n\\]\n\nThus $E_{n_{k,r}}-m$ is arbitrarily large, contradicting $E_{n}\\to m$.  \nTherefore the assumption was false and $b_{n}\\to\\infty$.  This proves (3a).\n\nStep 2.  Deduce (3b).  \nSince $b_{n}\\to\\infty$, the first term on the right–hand side of (8) tends to $0$; hence  \n\n\\[\n(n+m)\\Bigl(\\tfrac{b_{n+m}}{b_{n}}-1\\Bigr)\\longrightarrow 0 ,\n\\]\nwhich is exactly (3b).  Necessity is established.\n\nSufficiency.  \nConversely, if (3a)–(3b) hold, then $\\dfrac{m}{b_{n}}\\to 0$ and the last term in (4) also tends to $0$, so $E_{n}\\to m$. ∎\n\n\n\n────────────────────────────────────────  \nExamples of extremal sequences  \n────────────────────────────────────────  \n\n1. $a_{n}=n(\\log n)^{\\alpha}$, $\\alpha>0$.  \n   Then $L_{n}=(\\log n)^{\\alpha}\\to\\infty$, and a direct calculation gives  \n\\[\n(n+m)\\bigl(L_{n+m}/L_{n}-1\\bigr)\n      =\\frac{\\alpha m}{\\log n}+O\\!\\bigl(\\tfrac{1}{(\\log n)^{2}}\\bigr)\\longrightarrow 0.\n\\]\n\n2. $a_{n}=n\\exp\\!\\bigl[(\\log\\log n)^{\\beta}\\bigr]$, $0<\\beta<1$.  \n   Here $\\log L_{n}=(\\log\\log n)^{\\beta}$, and  \n\\[\n\\log\\!\\frac{L_{n+m}}{L_{n}}\n      =(\\log\\log(n+m))^{\\beta}-(\\log\\log n)^{\\beta}\n      =O\\!\\Bigl(\\frac{1}{n(\\log n)^{1-\\beta}}\\Bigr),\n\\]\nso $(3)$ is satisfied.\n\n3. More generally, any $a_{n}=nL_{n}$ with $L_{n}$ slowly varying (with step $m$) and $L_{n}\\to\\infty$ fulfils $E_{n}\\to m$, and every sequence with this property is of that form. ∎",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.450401",
        "was_fixed": false,
        "difficulty_analysis": "• Higher-order interaction:   \n  The expression now links the term aₙ to a_{n+m} across an arbitrary positive\n  offset m.  The proof must control the evolution of the sequence in jumps of\n  length m, rather than in consecutive terms.\n\n• Divergent series with stride m:  \n  Deriving the contradiction requires building an m-step harmonic series\n  ∑_{j}1/(k+jm), not the ordinary harmonic series that sufficed in the\n  original problem.  Handling this necessitates a careful index-shifting\n  argument and a non-trivial comparison with the usual ∑1/ℓ.\n\n• Parameter dependence:  \n  Both the inequality and the optimal constant depend explicitly on m.  One\n  must show simultaneously that  \n  (a) the lower bound scales exactly with m, and  \n  (b) the bound is tight for every m, by constructing appropriate families of\n  test sequences.  This introduces an additional layer of quantification and\n  verification absent from the original kernel.\n\n• More intricate recursion:  \n  Instead of a single forward estimate, the proof demands an m-step recursive\n  inequality and its iteration, leading to a telescoping sum with variable\n  denominators.  Careful bookkeeping of these denominators is essential to\n  guarantee divergence.\n\nOverall, the enhanced variant forces competitors to generalise the harmonic-series\nargument, manage non-unit step sizes, and track a parameter that permeates every\nstage of the reasoning, making the problem substantially more technical and\nconceptually demanding than the original."
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}