{
  "index": "1964-A-5",
  "type": "ANA",
  "tag": [
    "ANA",
    "ALG"
  ],
  "difficulty": "",
  "question": "5. Prove that there is a constant \\( K \\) such that the following inequality holds for any sequence of positive numbers \\( a_{1}, a_{2}, a_{3}, \\ldots \\) :\n\\[\n\\sum_{n=1}^{\\infty} \\frac{n}{a_{1}+a_{2}+\\cdots+a_{n}} \\leq K \\sum_{n=1}^{\\infty} \\frac{1}{a_{n}} . \\quad(\\text { page 589) }\n\\]",
  "solution": "Solution. Let \\( k=2 t \\) be some fixed even positive integer. We shall prove\n\\[\n\\sum_{n=1}^{k} \\frac{n}{a_{1}+a_{2}+\\cdots+a_{n}} \\leq 4 \\sum_{n=1}^{k} \\frac{1}{a_{n}}\n\\]\n\nFrom this inequality, the required inequality for infinite sums, with \\( K= \\) 4, follows immediately.\n\nLet \\( b_{1}, b_{2}, \\ldots, b_{k} \\) be the terms \\( a_{1}, a_{2}, \\ldots, a_{k} \\) enumerated in increasing order. For \\( 1 \\leq p \\leq t \\) we have\n\\[\n\\begin{aligned}\na_{1}+a_{2}+\\cdots+a_{2 p} & \\geq a_{1}+a_{2}+\\cdots+a_{2 p-1} \\\\\n& \\geq b_{1}+b_{2}+\\cdots+b_{2 p-1} \\geq p b_{p}\n\\end{aligned}\n\\]\nsince all terms are positive and since the last \\( p \\) terms are at least \\( b_{p} \\). Therefore\n\\[\n\\frac{2 p-1}{a_{1}+a_{2}+\\cdots+a_{2 p-1}} \\leq \\frac{2 p-1}{p b_{p}}<\\frac{2}{b_{p}} .\n\\]\n\nAlso\n\\[\n\\frac{2 p}{a_{1}+a_{2}+\\cdots+a_{2 p}} \\leq \\frac{2}{b_{p}} .\n\\]\n\nHence\n\\[\n\\frac{2 p-1}{a_{1}+a_{2}+\\cdots+a_{2 p-1}}+\\frac{2 p}{a_{1}+a_{2}+\\cdots+a_{2 p}}<\\frac{4}{b_{p}} .\n\\]\n\nThus\n\\[\n\\sum_{p=1}^{\\prime}\\left[\\frac{2 p-1}{a_{1}+a_{2}+\\cdots+a_{2 p-1}}+\\frac{2 p}{a_{1}+a_{2}+\\cdots+a_{2 p}}\\right]<\\sum_{p=1}^{\\prime} \\frac{4}{b_{p}} .\n\\]\n\nRewriting the left-hand sum, we have\n\\[\n\\sum_{n=1}^{k} \\frac{n}{a_{1}+a_{2}+\\cdots+a_{n}} \\leq \\sum_{p=1}^{\\prime} \\frac{4}{b_{p}}<\\sum_{p=1}^{k} \\frac{4}{b_{p}}=4 \\sum_{n=1}^{k} \\frac{1}{a_{n}} .\n\\]\n\nIf the series \\( \\sum_{n=1}^{\\infty}\\left(1 / a_{n}\\right) \\) diverges, the given inequality is deemed to be satisfied automatically. Otherwise we have\n\\[\n\\sum_{n=1}^{k} \\frac{n}{a_{1}+\\cdots+a_{n}} \\leq 4 \\sum_{n=1}^{\\infty} \\frac{1}{a_{n}}\n\\]\nand hence\n\\[\n\\sum_{n=1}^{\\infty} \\frac{n}{a_{1}+\\cdots+a_{n}} \\leq 4 \\sum_{n=1}^{\\infty} \\frac{1}{a_{n}} .\n\\]\n\nRemark. This inequality was first established by K. Knopp (Jour. London Math. Soc., vol 3 (1928), pp. 205-211). He also proved that the least constant \\( K \\) that will do is 2 .\n\nInspired by this problem, R. M. Redheffer (Proc. London Math. Soc., vol. 17 (1967), pp. 683-699) has investigated a large class of related inequalities. He proves among other things the following inequality that is stronger than Knopp's:\n\\[\n\\frac{3}{a_{1}}+\\frac{5}{a_{1}+a_{2}}+\\frac{7}{a_{1}+a_{2}+a_{3}}+\\cdots \\leq 4 \\Sigma \\frac{1}{a_{n}}\n\\]\n\nIn fact he proves the sharp finite version\n\\[\n\\frac{(k+1)^{2}}{A_{k}}+\\sum_{n=1}^{k-1} \\frac{2 n+1}{A_{n}} \\leq 4 \\sum_{n=1}^{k} \\frac{1}{a_{n}}\n\\]\nwhere \\( A_{n}=a_{1}+a_{2}+\\cdots+a_{n} \\), for any positive sequence \\( \\left\\{a_{n}\\right\\} \\) with equality if and only if \\( a_{1}, a_{2}, \\ldots, a_{k} \\) are proportional to \\( 1,2, \\ldots, k \\). Here is his proof of (2).\n\nLemma. Suppose \\( \\lambda>0 \\). Then\n\\[\n\\frac{(\\lambda+2)^{2}}{1+x} \\leq \\frac{4}{x}+\\lambda^{2}\n\\]\nfor all positive \\( x \\) with equality if and only if \\( x=2 / \\lambda \\).\nProof. Let\n\\[\nf(x)=\\frac{(\\lambda+2)^{2}}{1+x}-\\frac{4}{x}\n\\]\n\nThen\n\\[\nf^{\\prime}(x)=-\\frac{(\\lambda+2)^{2}}{(1+x)^{2}}+\\frac{4}{x^{2}}\n\\]\nand this is zero on \\( (0, \\infty) \\) only for \\( x=2 / \\lambda \\). This critical point is easily checked to be a maximum point for \\( f \\) and the lemma follows.\n\nProof of (2). This is clearly true for \\( k=1 \\). Assume it is true for \\( k=p \\).\nPutting \\( x=a_{p+1} / A_{p} \\) and applying the lemma with \\( \\lambda=p \\), we have\n\\[\n\\frac{(p+2)^{2}}{A_{p+1}}=\\frac{1}{A_{p}}\\left(\\frac{(p+2)^{2}}{1+x}\\right) \\leq \\frac{1}{A_{p}}\\left(\\frac{4}{x}+p^{2}\\right)=\\frac{4}{a_{p+1}}+\\frac{p^{2}}{A_{p}} .\n\\]\n\nAdding this inequality to the assumed inequality for \\( k=p \\), we get\n\\[\n\\frac{(p+2)^{2}}{A_{p+1}}+\\frac{(p+1)^{2}}{A_{p}}+\\sum_{n=1}^{p-1} \\frac{2 n+1}{A_{n}} \\leq 4 \\sum_{n=1}^{p+1} \\frac{1}{a_{n}}+\\frac{p^{2}}{A_{p}} .\n\\]\n\nCanceling \\( p^{2} / A_{p} \\) from both sides, we obtain (2) for \\( k=p+1 \\). This completes the proof by induction. There can be equality only if there is equality in (3) at each stage, i.e.. only if \\( a_{p+1}=2 A_{p} / p \\) from which follows that \\( a_{p} \\) \\( =p a_{1} \\).\n\nNow (1) follows immediately from (2). If the series on the right of (1) is convergent, then the \\( a \\) 's are not proportional to the integers. and it follows that the inequality is strict.",
  "vars": [
    "n",
    "k",
    "t",
    "p",
    "x",
    "f"
  ],
  "params": [
    "K",
    "a_n",
    "b_p",
    "A_n",
    "\\\\lambda"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "n": "indexer",
        "k": "bounder",
        "t": "halfsize",
        "p": "iterator",
        "x": "variable",
        "f": "function",
        "K": "constant",
        "a_n": "seqterm",
        "b_p": "sortedp",
        "A_n": "partialsum",
        "\\\\lambda": "lambda"
      },
      "question": "5. Prove that there is a constant \\( constant \\) such that the following inequality holds for any sequence of positive numbers \\( a_{1}, a_{2}, a_{3}, \\ldots \\) :\n\\[\n\\sum_{indexer=1}^{\\infty} \\frac{indexer}{a_{1}+a_{2}+\\cdots+seqterm} \\leq constant \\sum_{indexer=1}^{\\infty} \\frac{1}{seqterm} . \\quad(\\text { page 589) }\n\\]",
      "solution": "Solution. Let \\( bounder=2 halfsize \\) be some fixed even positive integer. We shall prove\n\\[\n\\sum_{indexer=1}^{bounder} \\frac{indexer}{a_{1}+a_{2}+\\cdots+a_{indexer}} \\leq 4 \\sum_{indexer=1}^{bounder} \\frac{1}{a_{indexer}}\n\\]\n\nFrom this inequality, the required inequality for infinite sums, with \\( constant= \\) 4, follows immediately.\n\nLet \\( b_{1}, b_{2}, \\ldots, b_{bounder} \\) be the terms \\( a_{1}, a_{2}, \\ldots, a_{bounder} \\) enumerated in increasing order. For \\( 1 \\leq iterator \\leq halfsize \\) we have\n\\[\n\\begin{aligned}\na_{1}+a_{2}+\\cdots+a_{2 iterator} & \\geq a_{1}+a_{2}+\\cdots+a_{2 iterator-1} \\\\\n& \\geq b_{1}+b_{2}+\\cdots+b_{2 iterator-1} \\geq iterator b_{iterator}\n\\end{aligned}\n\\]\nsince all terms are positive and since the last \\( iterator \\) terms are at least \\( sortedp \\). Therefore\n\\[\n\\frac{2 iterator-1}{a_{1}+a_{2}+\\cdots+a_{2 iterator-1}} \\leq \\frac{2 iterator-1}{iterator sortedp}<\\frac{2}{sortedp} .\n\\]\n\nAlso\n\\[\n\\frac{2 iterator}{a_{1}+a_{2}+\\cdots+a_{2 iterator}} \\leq \\frac{2}{sortedp} .\n\\]\n\nHence\n\\[\n\\frac{2 iterator-1}{a_{1}+a_{2}+\\cdots+a_{2 iterator-1}}+\\frac{2 iterator}{a_{1}+a_{2}+\\cdots+a_{2 iterator}}<\\frac{4}{sortedp} .\n\\]\n\nThus\n\\[\n\\sum_{iterator=1}^{\\prime}\\left[\\frac{2 iterator-1}{a_{1}+a_{2}+\\cdots+a_{2 iterator-1}}+\\frac{2 iterator}{a_{1}+a_{2}+\\cdots+a_{2 iterator}}\\right]<\\sum_{iterator=1}^{\\prime} \\frac{4}{sortedp} .\n\\]\n\nRewriting the left-hand sum, we have\n\\[\n\\sum_{indexer=1}^{bounder} \\frac{indexer}{a_{1}+a_{2}+\\cdots+a_{indexer}} \\leq \\sum_{iterator=1}^{\\prime} \\frac{4}{b_{iterator}}<\\sum_{iterator=1}^{bounder} \\frac{4}{b_{iterator}}=4 \\sum_{indexer=1}^{bounder} \\frac{1}{a_{indexer}} .\n\\]\n\nIf the series \\( \\sum_{indexer=1}^{\\infty}\\left(1 / seqterm\\right) \\) diverges, the given inequality is deemed to be satisfied automatically. Otherwise we have\n\\[\n\\sum_{indexer=1}^{bounder} \\frac{indexer}{a_{1}+\\cdots+a_{indexer}} \\leq 4 \\sum_{indexer=1}^{\\infty} \\frac{1}{seqterm}\n\\]\nand hence\n\\[\n\\sum_{indexer=1}^{\\infty} \\frac{indexer}{a_{1}+\\cdots+a_{indexer}} \\leq 4 \\sum_{indexer=1}^{\\infty} \\frac{1}{seqterm} .\n\\]\n\nRemark. This inequality was first established by K. Knopp (Jour. London Math. Soc., vol 3 (1928), pp. 205-211). He also proved that the least constant \\( constant \\) that will do is 2 .\n\nInspired by this problem, R. M. Redheffer (Proc. London Math. Soc., vol. 17 (1967), pp. 683-699) has investigated a large class of related inequalities. He proves among other things the following inequality that is stronger than Knopp's:\n\\[\n\\frac{3}{a_{1}}+\\frac{5}{a_{1}+a_{2}}+\\frac{7}{a_{1}+a_{2}+a_{3}}+\\cdots \\leq 4 \\Sigma \\frac{1}{seqterm}\n\\]\n\nIn fact he proves the sharp finite version\n\\[\n\\frac{(bounder+1)^{2}}{A_{bounder}}+\\sum_{indexer=1}^{bounder-1} \\frac{2 indexer+1}{A_{indexer}} \\leq 4 \\sum_{indexer=1}^{bounder} \\frac{1}{a_{indexer}}\n\\]\nwhere \\( partialsum=a_{1}+a_{2}+\\cdots+a_{indexer} \\), for any positive sequence \\( \\left\\{seqterm\\right\\} \\) with equality if and only if \\( a_{1}, a_{2}, \\ldots, a_{bounder} \\) are proportional to \\( 1,2, \\ldots, bounder \\). Here is his proof of (2).\n\nLemma. Suppose \\( lambda>0 \\). Then\n\\[\n\\frac{(lambda+2)^{2}}{1+variable} \\leq \\frac{4}{variable}+lambda^{2}\n\\]\nfor all positive \\( variable \\) with equality if and only if \\( variable=2 / lambda \\).\nProof. Let\n\\[\nfunction(variable)=\\frac{(lambda+2)^{2}}{1+variable}-\\frac{4}{variable}\n\\]\n\nThen\n\\[\nfunction^{\\prime}(variable)=-\\frac{(lambda+2)^{2}}{(1+variable)^{2}}+\\frac{4}{variable^{2}}\n\\]\nand this is zero on \\( (0, \\infty) \\) only for \\( variable=2 / lambda \\). This critical point is easily checked to be a maximum point for \\( function \\) and the lemma follows.\n\nProof of (2). This is clearly true for \\( bounder=1 \\). Assume it is true for \\( bounder=iterator \\).\nPutting \\( variable=a_{iterator+1} / A_{iterator} \\) and applying the lemma with \\( lambda=iterator \\), we have\n\\[\n\\frac{(iterator+2)^{2}}{A_{iterator+1}}=\\frac{1}{A_{iterator}}\\left(\\frac{(iterator+2)^{2}}{1+variable}\\right) \\leq \\frac{1}{A_{iterator}}\\left(\\frac{4}{variable}+iterator^{2}\\right)=\\frac{4}{a_{iterator+1}}+\\frac{iterator^{2}}{A_{iterator}} .\n\\]\n\nAdding this inequality to the assumed inequality for \\( bounder=iterator \\), we get\n\\[\n\\frac{(iterator+2)^{2}}{A_{iterator+1}}+\\frac{(iterator+1)^{2}}{A_{iterator}}+\\sum_{indexer=1}^{iterator-1} \\frac{2 indexer+1}{A_{indexer}} \\leq 4 \\sum_{indexer=1}^{iterator+1} \\frac{1}{a_{indexer}}+\\frac{iterator^{2}}{A_{iterator}} .\n\\]\n\nCanceling \\( iterator^{2} / A_{iterator} \\) from both sides, we obtain (2) for \\( bounder=iterator+1 \\). This completes the proof by induction. There can be equality only if there is equality in (3) at each stage, i.e., only if \\( a_{iterator+1}=2 A_{iterator} / iterator \\) from which follows that \\( a_{iterator} \\) \\( =iterator a_{1} \\).\n\nNow (1) follows immediately from (2). If the series on the right of (1) is convergent, then the \\( a \\) 's are not proportional to the integers, and it follows that the inequality is strict."
    },
    "descriptive_long_confusing": {
      "map": {
        "n": "sandstone",
        "k": "whirlwind",
        "t": "blueberry",
        "p": "firebrand",
        "x": "blackbird",
        "f": "riverbank",
        "K": "mountaine",
        "a_n": "waterfall",
        "b_p": "cinnamon",
        "A_n": "evergreen",
        "\\\\lambda": "hummingbird"
      },
      "question": "5. Prove that there is a constant \\( mountaine \\) such that the following inequality holds for any sequence of positive numbers \\( waterfall_{1}, waterfall_{2}, waterfall_{3}, \\ldots \\) :\n\\[\n\\sum_{sandstone=1}^{\\infty} \\frac{sandstone}{waterfall_{1}+waterfall_{2}+\\cdots+waterfall_{sandstone}} \\leq mountaine \\sum_{sandstone=1}^{\\infty} \\frac{1}{waterfall_{sandstone}} . \\quad(\\text { page 589) }\n\\]",
      "solution": "Solution. Let \\( whirlwind=2 blueberry \\) be some fixed even positive integer. We shall prove\n\\[\n\\sum_{sandstone=1}^{whirlwind} \\frac{sandstone}{waterfall_{1}+waterfall_{2}+\\cdots+waterfall_{sandstone}} \\leq 4 \\sum_{sandstone=1}^{whirlwind} \\frac{1}{waterfall_{sandstone}}\n\\]\n\nFrom this inequality, the required inequality for infinite sums, with \\( mountaine= \\) 4, follows immediately.\n\nLet \\( cinnamon_{1}, cinnamon_{2}, \\ldots, cinnamon_{whirlwind} \\) be the terms \\( waterfall_{1}, waterfall_{2}, \\ldots, waterfall_{whirlwind} \\) enumerated in increasing order. For \\( 1 \\leq firebrand \\leq blueberry \\) we have\n\\[\n\\begin{aligned}\nwaterfall_{1}+waterfall_{2}+\\cdots+waterfall_{2 firebrand} & \\geq waterfall_{1}+waterfall_{2}+\\cdots+waterfall_{2 firebrand-1} \\\\\n& \\geq cinnamon_{1}+cinnamon_{2}+\\cdots+cinnamon_{2 firebrand-1} \\geq firebrand\\, cinnamon_{firebrand}\n\\end{aligned}\n\\]\nsince all terms are positive and since the last \\( firebrand \\) terms are at least \\( cinnamon_{firebrand} \\). Therefore\n\\[\n\\frac{2 firebrand-1}{waterfall_{1}+waterfall_{2}+\\cdots+waterfall_{2 firebrand-1}} \\leq \\frac{2 firebrand-1}{firebrand\\, cinnamon_{firebrand}}<\\frac{2}{cinnamon_{firebrand}} .\n\\]\n\nAlso\n\\[\n\\frac{2 firebrand}{waterfall_{1}+waterfall_{2}+\\cdots+waterfall_{2 firebrand}} \\leq \\frac{2}{cinnamon_{firebrand}} .\n\\]\n\nHence\n\\[\n\\frac{2 firebrand-1}{waterfall_{1}+waterfall_{2}+\\cdots+waterfall_{2 firebrand-1}}+\\frac{2 firebrand}{waterfall_{1}+waterfall_{2}+\\cdots+waterfall_{2 firebrand}}<\\frac{4}{cinnamon_{firebrand}} .\n\\]\n\nThus\n\\[\n\\sum_{firebrand=1}^{\\prime}\\left[\\frac{2 firebrand-1}{waterfall_{1}+waterfall_{2}+\\cdots+waterfall_{2 firebrand-1}}+\\frac{2 firebrand}{waterfall_{1}+waterfall_{2}+\\cdots+waterfall_{2 firebrand}}\\right]<\\sum_{firebrand=1}^{\\prime} \\frac{4}{cinnamon_{firebrand}} .\n\\]\n\nRewriting the left-hand sum, we have\n\\[\n\\sum_{sandstone=1}^{whirlwind} \\frac{sandstone}{waterfall_{1}+waterfall_{2}+\\cdots+waterfall_{sandstone}} \\leq \\sum_{firebrand=1}^{\\prime} \\frac{4}{cinnamon_{firebrand}}<\\sum_{firebrand=1}^{whirlwind} \\frac{4}{cinnamon_{firebrand}}=4 \\sum_{sandstone=1}^{whirlwind} \\frac{1}{waterfall_{sandstone}} .\n\\]\n\nIf the series \\( \\sum_{sandstone=1}^{\\infty}\\left(1 / waterfall_{sandstone}\\right) \\) diverges, the given inequality is deemed to be satisfied automatically. Otherwise we have\n\\[\n\\sum_{sandstone=1}^{whirlwind} \\frac{sandstone}{waterfall_{1}+\\cdots+waterfall_{sandstone}} \\leq 4 \\sum_{sandstone=1}^{\\infty} \\frac{1}{waterfall_{sandstone}}\n\\]\nand hence\n\\[\n\\sum_{sandstone=1}^{\\infty} \\frac{sandstone}{waterfall_{1}+\\cdots+waterfall_{sandstone}} \\leq 4 \\sum_{sandstone=1}^{\\infty} \\frac{1}{waterfall_{sandstone}} .\n\\]\n\nRemark. This inequality was first established by K. Knopp (Jour. London Math. Soc., vol 3 (1928), pp. 205-211). He also proved that the least constant \\( mountaine \\) that will do is 2 .\n\nInspired by this problem, R. M. Redheffer (Proc. London Math. Soc., vol. 17 (1967), pp. 683-699) has investigated a large class of related inequalities. He proves among other things the following inequality that is stronger than Knopp's:\n\\[\n\\frac{3}{waterfall_{1}}+\\frac{5}{waterfall_{1}+waterfall_{2}}+\\frac{7}{waterfall_{1}+waterfall_{2}+waterfall_{3}}+\\cdots \\leq 4 \\Sigma \\frac{1}{waterfall_{sandstone}}\n\\]\n\nIn fact he proves the sharp finite version\n\\[\n\\frac{(whirlwind+1)^{2}}{evergreen_{whirlwind}}+\\sum_{sandstone=1}^{whirlwind-1} \\frac{2 sandstone+1}{evergreen_{sandstone}} \\leq 4 \\sum_{sandstone=1}^{whirlwind} \\frac{1}{waterfall_{sandstone}}\n\\]\nwhere \\( evergreen_{sandstone}=waterfall_{1}+waterfall_{2}+\\cdots+waterfall_{sandstone} \\), for any positive sequence \\( \\left\\{waterfall_{sandstone}\\right\\} \\) with equality if and only if \\( waterfall_{1}, waterfall_{2}, \\ldots, waterfall_{whirlwind} \\) are proportional to \\( 1,2, \\ldots, whirlwind \\). Here is his proof of (2).\n\nLemma. Suppose \\( hummingbird>0 \\). Then\n\\[\n\\frac{(hummingbird+2)^{2}}{1+blackbird} \\leq \\frac{4}{blackbird}+hummingbird^{2}\n\\]\nfor all positive \\( blackbird \\) with equality if and only if \\( blackbird=2 / hummingbird \\).\nProof. Let\n\\[\nriverbank(blackbird)=\\frac{(hummingbird+2)^{2}}{1+blackbird}-\\frac{4}{blackbird}\n\\]\n\nThen\n\\[\nriverbank^{\\prime}(blackbird)=-\\frac{(hummingbird+2)^{2}}{(1+blackbird)^{2}}+\\frac{4}{blackbird^{2}}\n\\]\nand this is zero on \\( (0, \\infty) \\) only for \\( blackbird=2 / hummingbird \\). This critical point is easily checked to be a maximum point for \\( riverbank \\) and the lemma follows.\n\nProof of (2). This is clearly true for \\( whirlwind=1 \\). Assume it is true for \\( whirlwind=firebrand \\).\nPutting \\( blackbird=waterfall_{firebrand+1} / evergreen_{firebrand} \\) and applying the lemma with \\( hummingbird=firebrand \\), we have\n\\[\n\\frac{(firebrand+2)^{2}}{evergreen_{firebrand+1}}=\\frac{1}{evergreen_{firebrand}}\\left(\\frac{(firebrand+2)^{2}}{1+blackbird}\\right) \\leq \\frac{1}{evergreen_{firebrand}}\\left(\\frac{4}{blackbird}+firebrand^{2}\\right)=\\frac{4}{waterfall_{firebrand+1}}+\\frac{firebrand^{2}}{evergreen_{firebrand}} .\n\\]\n\nAdding this inequality to the assumed inequality for \\( whirlwind=firebrand \\), we get\n\\[\n\\frac{(firebrand+2)^{2}}{evergreen_{firebrand+1}}+\\frac{(firebrand+1)^{2}}{evergreen_{firebrand}}+\\sum_{sandstone=1}^{firebrand-1} \\frac{2 sandstone+1}{evergreen_{sandstone}} \\leq 4 \\sum_{sandstone=1}^{firebrand+1} \\frac{1}{waterfall_{sandstone}}+\\frac{firebrand^{2}}{evergreen_{firebrand}} .\n\\]\n\nCanceling \\( firebrand^{2} / evergreen_{firebrand} \\) from both sides, we obtain (2) for \\( whirlwind=firebrand+1 \\). This completes the proof by induction. There can be equality only if there is equality in (3) at each stage, i.e., only if \\( waterfall_{firebrand+1}=2 evergreen_{firebrand} / firebrand \\) from which follows that \\( waterfall_{firebrand}=firebrand\\, waterfall_{1} \\).\n\nNow (1) follows immediately from (2). If the series on the right of (1) is convergent, then the waterfalls are not proportional to the integers, and it follows that the inequality is strict."
    },
    "descriptive_long_misleading": {
      "map": {
        "n": "continuum",
        "k": "fractional",
        "t": "spaceunit",
        "p": "composite",
        "x": "knownvalue",
        "f": "constant",
        "K": "variable",
        "a_n": "negativeseq",
        "b_p": "unorderedseq",
        "A_n": "difference",
        "\\\\lambda": "infinite"
      },
      "question": "5. Prove that there is a constant \\( variable \\) such that the following inequality holds for any sequence of positive numbers \\( negativeseq_{1}, negativeseq_{2}, negativeseq_{3}, \\ldots \\) :\n\\[\n\\sum_{continuum=1}^{\\infty} \\frac{continuum}{negativeseq_{1}+negativeseq_{2}+\\cdots+negativeseq_{continuum}} \\leq variable \\sum_{continuum=1}^{\\infty} \\frac{1}{negativeseq_{continuum}} . \\quad(\\text { page 589) }\n\\]\n",
      "solution": "Solution. Let \\( fractional=2 spaceunit \\) be some fixed even positive integer. We shall prove\n\\[\n\\sum_{continuum=1}^{fractional} \\frac{continuum}{negativeseq_{1}+negativeseq_{2}+\\cdots+negativeseq_{continuum}} \\leq 4 \\sum_{continuum=1}^{fractional} \\frac{1}{negativeseq_{continuum}}\n\\]\n\nFrom this inequality, the required inequality for infinite sums, with \\( K= \\) 4, follows immediately.\n\nLet \\( unorderedseq_{1}, unorderedseq_{2}, \\ldots, unorderedseq_{fractional} \\) be the terms \\( negativeseq_{1}, negativeseq_{2}, \\ldots, negativeseq_{fractional} \\) enumerated in increasing order. For \\( 1 \\leq composite \\leq spaceunit \\) we have\n\\[\n\\begin{aligned}\nnegativeseq_{1}+negativeseq_{2}+\\cdots+negativeseq_{2 composite} & \\geq negativeseq_{1}+negativeseq_{2}+\\cdots+negativeseq_{2 composite-1} \\\\\n& \\geq unorderedseq_{1}+unorderedseq_{2}+\\cdots+unorderedseq_{2 composite-1} \\geq composite\\, unorderedseq_{composite}\n\\end{aligned}\n\\]\nsince all terms are positive and since the last \\( composite \\) terms are at least \\( unorderedseq_{composite} \\). Therefore\n\\[\n\\frac{2 composite-1}{negativeseq_{1}+negativeseq_{2}+\\cdots+negativeseq_{2 composite-1}} \\leq \\frac{2 composite-1}{composite\\, unorderedseq_{composite}}<\\frac{2}{unorderedseq_{composite}} .\n\\]\n\nAlso\n\\[\n\\frac{2 composite}{negativeseq_{1}+negativeseq_{2}+\\cdots+negativeseq_{2 composite}} \\leq \\frac{2}{unorderedseq_{composite}} .\n\\]\n\nHence\n\\[\n\\frac{2 composite-1}{negativeseq_{1}+negativeseq_{2}+\\cdots+negativeseq_{2 composite-1}}+\\frac{2 composite}{negativeseq_{1}+negativeseq_{2}+\\cdots+negativeseq_{2 composite}}<\\frac{4}{unorderedseq_{composite}} .\n\\]\n\nThus\n\\[\n\\sum_{composite=1}^{\\prime}\\left[\\frac{2 composite-1}{negativeseq_{1}+negativeseq_{2}+\\cdots+negativeseq_{2 composite-1}}+\\frac{2 composite}{negativeseq_{1}+negativeseq_{2}+\\cdots+negativeseq_{2 composite}}\\right]<\\sum_{composite=1}^{\\prime} \\frac{4}{unorderedseq_{composite}} .\n\\]\n\nRewriting the left-hand sum, we have\n\\[\n\\sum_{continuum=1}^{fractional} \\frac{continuum}{negativeseq_{1}+negativeseq_{2}+\\cdots+negativeseq_{continuum}} \\leq \\sum_{composite=1}^{\\prime} \\frac{4}{unorderedseq_{composite}}<\\sum_{composite=1}^{fractional} \\frac{4}{unorderedseq_{composite}}=4 \\sum_{continuum=1}^{fractional} \\frac{1}{negativeseq_{continuum}} .\n\\]\n\nIf the series \\( \\sum_{continuum=1}^{\\infty}\\left(1 / negativeseq_{continuum}\\right) \\) diverges, the given inequality is deemed to be satisfied automatically. Otherwise we have\n\\[\n\\sum_{continuum=1}^{fractional} \\frac{continuum}{negativeseq_{1}+\\cdots+negativeseq_{continuum}} \\leq 4 \\sum_{continuum=1}^{\\infty} \\frac{1}{negativeseq_{continuum}}\n\\]\nand hence\n\\[\n\\sum_{continuum=1}^{\\infty} \\frac{continuum}{negativeseq_{1}+\\cdots+negativeseq_{continuum}} \\leq 4 \\sum_{continuum=1}^{\\infty} \\frac{1}{negativeseq_{continuum}} .\n\\]\n\nRemark. This inequality was first established by K. Knopp (Jour. London Math. Soc., vol 3 (1928), pp. 205-211). He also proved that the least constant \\( K \\) that will do is 2 .\n\nInspired by this problem, R. M. Redheffer (Proc. London Math. Soc., vol. 17 (1967), pp. 683-699) has investigated a large class of related inequalities. He proves among other things the following inequality that is stronger than Knopp's:\n\\[\n\\frac{3}{negativeseq_{1}}+\\frac{5}{negativeseq_{1}+negativeseq_{2}}+\\frac{7}{negativeseq_{1}+negativeseq_{2}+negativeseq_{3}}+\\cdots \\leq 4 \\Sigma \\frac{1}{negativeseq_{continuum}}\n\\]\n\nIn fact he proves the sharp finite version\n\\[\n\\frac{(fractional+1)^{2}}{difference_{fractional}}+\\sum_{continuum=1}^{fractional-1} \\frac{2 continuum+1}{difference_{continuum}} \\leq 4 \\sum_{continuum=1}^{fractional} \\frac{1}{negativeseq_{continuum}}\n\\]\nwhere \\( difference_{continuum}=negativeseq_{1}+negativeseq_{2}+\\cdots+negativeseq_{continuum} \\), for any positive sequence \\( \\left\\{negativeseq_{continuum}\\right\\} \\) with equality if and only if \\( negativeseq_{1}, negativeseq_{2}, \\ldots, negativeseq_{fractional} \\) are proportional to \\( 1,2, \\ldots, fractional \\). Here is his proof of (2).\n\nLemma. Suppose \\( infinite>0 \\). Then\n\\[\n\\frac{(infinite+2)^{2}}{1+knownvalue} \\leq \\frac{4}{knownvalue}+infinite^{2}\n\\]\nfor all positive \\( knownvalue \\) with equality if and only if \\( knownvalue=2 / infinite \\).\nProof. Let\n\\[\nconstant(knownvalue)=\\frac{(infinite+2)^{2}}{1+knownvalue}-\\frac{4}{knownvalue}\n\\]\n\nThen\n\\[\nconstant^{\\prime}(knownvalue)=-\\frac{(infinite+2)^{2}}{(1+knownvalue)^{2}}+\\frac{4}{knownvalue^{2}}\n\\]\nand this is zero on \\( (0, \\infty) \\) only for \\( knownvalue=2 / infinite \\). This critical point is easily checked to be a maximum point for \\( constant \\) and the lemma follows.\n\nProof of (2). This is clearly true for \\( fractional=1 \\). Assume it is true for \\( fractional=composite \\).\nPutting \\( knownvalue=negativeseq_{composite+1} / difference_{composite} \\) and applying the lemma with \\( infinite=composite \\), we have\n\\[\n\\frac{(composite+2)^{2}}{difference_{composite+1}}=\\frac{1}{difference_{composite}}\\left(\\frac{(composite+2)^{2}}{1+knownvalue}\\right) \\leq \\frac{1}{difference_{composite}}\\left(\\frac{4}{knownvalue}+composite^{2}\\right)=\\frac{4}{negativeseq_{composite+1}}+\\frac{composite^{2}}{difference_{composite}} .\n\\]\n\nAdding this inequality to the assumed inequality for \\( fractional=composite \\), we get\n\\[\n\\frac{(composite+2)^{2}}{difference_{composite+1}}+\\frac{(composite+1)^{2}}{difference_{composite}}+\\sum_{continuum=1}^{composite-1} \\frac{2 continuum+1}{difference_{continuum}} \\leq 4 \\sum_{continuum=1}^{composite+1} \\frac{1}{negativeseq_{continuum}}+\\frac{composite^{2}}{difference_{composite}} .\n\\]\n\nCanceling \\( composite^{2} / difference_{composite} \\) from both sides, we obtain (2) for \\( fractional=composite+1 \\). This completes the proof by induction. There can be equality only if there is equality in (3) at each stage, i.e.. only if \\( negativeseq_{composite+1}=2 difference_{composite} / composite \\) from which follows that \\( negativeseq_{composite}=composite negativeseq_{1} \\).\n\nNow (1) follows immediately from (2). If the series on the right of (1) is convergent, then the negativeseq 's are not proportional to the integers, and it follows that the inequality is strict.\n"
    },
    "garbled_string": {
      "map": {
        "n": "qzxwvtnp",
        "k": "hjgrksla",
        "t": "mndxpsfo",
        "p": "ulvertnc",
        "x": "fzlkmqpa",
        "f": "wertyxcv",
        "K": "oskmnefg",
        "a_n": "jgklsdte",
        "b_p": "csplrofi",
        "A_n": "derewplk",
        "\\\\lambda": "xyqmnvab"
      },
      "question": "5. Prove that there is a constant \\( oskmnefg \\) such that the following inequality holds for any sequence of positive numbers \\( jgklsdte_{1}, jgklsdte_{2}, jgklsdte_{3}, \\ldots \\) :\n\\[\n\\sum_{qzxwvtnp=1}^{\\infty} \\frac{qzxwvtnp}{jgklsdte_{1}+jgklsdte_{2}+\\cdots+jgklsdte_{qzxwvtnp}} \\leq oskmnefg \\sum_{qzxwvtnp=1}^{\\infty} \\frac{1}{jgklsdte_{qzxwvtnp}} . \\quad(\\text { page 589) }\n\\]",
      "solution": "Solution. Let \\( hjgrksla=2 mndxpsfo \\) be some fixed even positive integer. We shall prove\n\\[\n\\sum_{qzxwvtnp=1}^{hjgrksla} \\frac{qzxwvtnp}{jgklsdte_{1}+jgklsdte_{2}+\\cdots+jgklsdte_{qzxwvtnp}} \\leq 4 \\sum_{qzxwvtnp=1}^{hjgrksla} \\frac{1}{jgklsdte_{qzxwvtnp}}\n\\]\n\nFrom this inequality, the required inequality for infinite sums, with \\( oskmnefg=4 \\), follows immediately.\n\nLet \\( csplrofi_{1}, csplrofi_{2}, \\ldots, csplrofi_{hjgrksla} \\) be the terms \\( jgklsdte_{1}, jgklsdte_{2}, \\ldots, jgklsdte_{hjgrksla} \\) enumerated in increasing order. For \\( 1 \\leq ulvertnc \\leq mndxpsfo \\) we have\n\\[\n\\begin{aligned}\njgklsdte_{1}+jgklsdte_{2}+\\cdots+jgklsdte_{2 ulvertnc}& \\geq jgklsdte_{1}+jgklsdte_{2}+\\cdots+jgklsdte_{2 ulvertnc-1}\\\\\n& \\geq csplrofi_{1}+csplrofi_{2}+\\cdots+csplrofi_{2 ulvertnc-1} \\geq ulvertnc\\,csplrofi_{ulvertnc}\n\\end{aligned}\n\\]\nsince all terms are positive and since the last \\( ulvertnc \\) terms are at least \\( csplrofi_{ulvertnc} \\). Therefore\n\\[\n\\frac{2 ulvertnc-1}{jgklsdte_{1}+jgklsdte_{2}+\\cdots+jgklsdte_{2 ulvertnc-1}} \\leq \\frac{2 ulvertnc-1}{ulvertnc\\,csplrofi_{ulvertnc}}<\\frac{2}{csplrofi_{ulvertnc}} .\n\\]\n\nAlso\n\\[\n\\frac{2 ulvertnc}{jgklsdte_{1}+jgklsdte_{2}+\\cdots+jgklsdte_{2 ulvertnc}} \\leq \\frac{2}{csplrofi_{ulvertnc}} .\n\\]\n\nHence\n\\[\n\\frac{2 ulvertnc-1}{jgklsdte_{1}+jgklsdte_{2}+\\cdots+jgklsdte_{2 ulvertnc-1}}+\\frac{2 ulvertnc}{jgklsdte_{1}+jgklsdte_{2}+\\cdots+jgklsdte_{2 ulvertnc}}<\\frac{4}{csplrofi_{ulvertnc}} .\n\\]\n\nThus\n\\[\n\\sum_{ulvertnc=1}^{\\prime}\\left[\\frac{2 ulvertnc-1}{jgklsdte_{1}+jgklsdte_{2}+\\cdots+jgklsdte_{2 ulvertnc-1}}+\\frac{2 ulvertnc}{jgklsdte_{1}+jgklsdte_{2}+\\cdots+jgklsdte_{2 ulvertnc}}\\right]<\\sum_{ulvertnc=1}^{\\prime} \\frac{4}{csplrofi_{ulvertnc}} .\n\\]\n\nRewriting the left-hand sum, we have\n\\[\n\\sum_{ulvertnc=1}^{hjgrksla} \\frac{ulvertnc}{jgklsdte_{1}+jgklsdte_{2}+\\cdots+jgklsdte_{ulvertnc}} \\leq \\sum_{ulvertnc=1}^{\\prime} \\frac{4}{csplrofi_{ulvertnc}}<\\sum_{ulvertnc=1}^{hjgrksla} \\frac{4}{csplrofi_{ulvertnc}}=4 \\sum_{qzxwvtnp=1}^{hjgrksla} \\frac{1}{jgklsdte_{qzxwvtnp}} .\n\\]\n\nIf the series \\( \\sum_{qzxwvtnp=1}^{\\infty}(1 / jgklsdte_{qzxwvtnp}) \\) diverges, the given inequality is deemed to be satisfied automatically. Otherwise we have\n\\[\n\\sum_{qzxwvtnp=1}^{hjgrksla} \\frac{qzxwvtnp}{jgklsdte_{1}+\\cdots+jgklsdte_{qzxwvtnp}} \\leq 4 \\sum_{qzxwvtnp=1}^{\\infty} \\frac{1}{jgklsdte_{qzxwvtnp}}\n\\]\nand hence\n\\[\n\\sum_{qzxwvtnp=1}^{\\infty} \\frac{qzxwvtnp}{jgklsdte_{1}+\\cdots+jgklsdte_{qzxwvtnp}} \\leq 4 \\sum_{qzxwvtnp=1}^{\\infty} \\frac{1}{jgklsdte_{qzxwvtnp}} .\n\\]\n\nRemark. This inequality was first established by K. Knopp (Jour. London Math. Soc., vol 3 (1928), pp. 205-211). He also proved that the least constant \\( oskmnefg \\) that will do is 2.\n\nInspired by this problem, R. M. Redheffer (Proc. London Math. Soc., vol. 17 (1967), pp. 683-699) has investigated a large class of related inequalities. He proves among other things the following inequality that is stronger than Knopp's:\n\\[\n\\frac{3}{jgklsdte_{1}}+\\frac{5}{jgklsdte_{1}+jgklsdte_{2}}+\\frac{7}{jgklsdte_{1}+jgklsdte_{2}+jgklsdte_{3}}+\\cdots \\leq 4 \\Sigma \\frac{1}{jgklsdte_{qzxwvtnp}}\n\\]\n\nIn fact he proves the sharp finite version\n\\[\n\\frac{(hjgrksla+1)^{2}}{derewplk_{hjgrksla}}+\\sum_{qzxwvtnp=1}^{hjgrksla-1} \\frac{2 qzxwvtnp+1}{derewplk_{qzxwvtnp}} \\leq 4 \\sum_{qzxwvtnp=1}^{hjgrksla} \\frac{1}{jgklsdte_{qzxwvtnp}}\n\\]\nwhere \\( derewplk_{qzxwvtnp}=jgklsdte_{1}+jgklsdte_{2}+\\cdots+jgklsdte_{qzxwvtnp} \\), for any positive sequence \\( \\{jgklsdte_{qzxwvtnp}\\} \\) with equality if and only if \\( jgklsdte_{1}, jgklsdte_{2}, \\ldots, jgklsdte_{hjgrksla} \\) are proportional to \\( 1,2, \\ldots, hjgrksla \\). Here is his proof of (2).\n\nLemma. Suppose \\( xyqmnvab>0 \\). Then\n\\[\n\\frac{(xyqmnvab+2)^{2}}{1+fzlkmqpa} \\leq \\frac{4}{fzlkmqpa}+xyqmnvab^{2}\n\\]\nfor all positive \\( fzlkmqpa \\) with equality if and only if \\( fzlkmqpa=2 / xyqmnvab \\).\n\nProof. Let\n\\[\nwertyxcv(fzlkmqpa)=\\frac{(xyqmnvab+2)^{2}}{1+fzlkmqpa}-\\frac{4}{fzlkmqpa}\n\\]\n\nThen\n\\[\nwertyxcv^{\\prime}(fzlkmqpa)=-\\frac{(xyqmnvab+2)^{2}}{(1+fzlkmqpa)^{2}}+\\frac{4}{fzlkmqpa^{2}}\n\\]\nand this is zero on \\( (0, \\infty) \\) only for \\( fzlkmqpa=2 / xyqmnvab \\). This critical point is easily checked to be a maximum point for \\( wertyxcv \\) and the lemma follows.\n\nProof of (2). This is clearly true for \\( hjgrksla=1 \\). Assume it is true for \\( hjgrksla=ulvertnc \\).\nPutting \\( fzlkmqpa=jgklsdte_{ulvertnc+1} / derewplk_{ulvertnc} \\) and applying the lemma with \\( xyqmnvab=ulvertnc \\), we have\n\\[\n\\frac{(ulvertnc+2)^{2}}{derewplk_{ulvertnc+1}}=\\frac{1}{derewplk_{ulvertnc}}\\left(\\frac{(ulvertnc+2)^{2}}{1+fzlkmqpa}\\right) \\leq \\frac{1}{derewplk_{ulvertnc}}\\left(\\frac{4}{fzlkmqpa}+ulvertnc^{2}\\right)=\\frac{4}{jgklsdte_{ulvertnc+1}}+\\frac{ulvertnc^{2}}{derewplk_{ulvertnc}} .\n\\]\n\nAdding this inequality to the assumed inequality for \\( hjgrksla=ulvertnc \\), we get\n\\[\n\\frac{(ulvertnc+2)^{2}}{derewplk_{ulvertnc+1}}+\\frac{(ulvertnc+1)^{2}}{derewplk_{ulvertnc}}+\\sum_{qzxwvtnp=1}^{ulvertnc-1} \\frac{2 qzxwvtnp+1}{derewplk_{qzxwvtnp}} \\leq 4 \\sum_{qzxwvtnp=1}^{ulvertnc+1} \\frac{1}{jgklsdte_{qzxwvtnp}}+\\frac{ulvertnc^{2}}{derewplk_{ulvertnc}} .\n\\]\n\nCanceling \\( ulvertnc^{2} / derewplk_{ulvertnc} \\) from both sides, we obtain (2) for \\( hjgrksla=ulvertnc+1 \\). This completes the proof by induction. There can be equality only if there is equality in (3) at each stage, i.e., only if \\( jgklsdte_{ulvertnc+1}=2 derewplk_{ulvertnc} / ulvertnc \\) from which follows that \\( jgklsdte_{ulvertnc}=ulvertnc\\,jgklsdte_{1} \\).\n\nNow (1) follows immediately from (2). If the series on the right of (1) is convergent, then the \\( jgklsdte \\)'s are not proportional to the integers, and it follows that the inequality is strict."
    },
    "kernel_variant": {
      "question": "Let $\\bigl(a_{n}\\bigr)_{n\\ge 1}$ be an arbitrary sequence of strictly positive real numbers and put  \n\\[\nA_{n}=a_{1}+a_{2}+\\dots +a_{n}\\qquad(n\\ge 1).\n\\]\n\na) (Knopp’s inequality – optimal form)  \nProve that  \n\\[\n\\boxed{\\;\n   \\displaystyle \n   \\sum_{n=1}^{\\infty}\\frac{n}{A_{n}}\\le 2\\sum_{n=1}^{\\infty}\\frac{1}{a_{n}}\n  \\;} \\tag{K}\n\\]\nand show that the numerical constant $2$ is best possible.\n\nb) (Impossibility of equality)  \nShow that if $\\displaystyle\\sum_{n=1}^{\\infty}1/a_{n}<\\infty$, then the\ninequality in (K) is always strict.  \n(When $\\sum 1/a_{n}=+\\infty$ the right–hand side of (K) is infinite,\nso equality is meaningless.)\n\nc) (Redheffer refinement – sharp finite and infinite versions)  \nFor every positive integer $k$ prove  \n\\[\n\\boxed{\\;\n   \\frac{(k+1)^{2}}{A_{k}}+\\sum_{n=1}^{k-1}\\frac{2n+1}{A_{n}}\n               \\le 4\\sum_{n=1}^{k}\\frac{1}{a_{n}}\n  \\;} \\tag{R}\n\\]\nand deduce the infinite counterpart  \n\\[\n\\boxed{\\;\n    \\sum_{n=1}^{\\infty}\\frac{2n+1}{A_{n}}\n           \\le 4\\sum_{n=1}^{\\infty}\\frac{1}{a_{n}}\n   \\;} \\tag{R_\\infty}\n\\]\nFinally, prove that $4$ is the best possible constant in both (R) and $(R_\\infty)$.\n\n\\vspace{1ex}",
      "solution": "Throughout write  \n\\[\ns_{n}:=\\frac{n}{A_{n}},\\qquad   \n\\Delta_{n}:=\\frac{1}{a_{n}},\\qquad\nB_{n}:=\\sum_{j=1}^{n}\\Delta_{j}\\ (n\\ge 1),\\qquad \nB_{\\infty}:=\\sum_{j\\ge 1}\\Delta_{j}.\n\\]\n\n%------------------------------------------------------------\n\\textbf{Part (c) – Redheffer’s refinement with the sharp constant $4$}\n%------------------------------------------------------------\n(The argument is self–contained and will later be used to settle parts\n(a) and (b).)\n\n\\textit{Lemma (Redheffer).}  \nFor every $x>0$ and $\\lambda>0$ one has  \n\\[\n\\frac{(\\lambda+2)^{2}}{1+x}\\le\\frac{4}{x}+\\lambda^{2},\n\\qquad\\text{with equality iff }x=\\frac{2}{\\lambda}.\n\\]\n\n\\textit{Proof.}  \nPut $f(x)=\\dfrac{(\\lambda+2)^{2}}{1+x}-\\dfrac{4}{x}$.  \nA routine derivative computation gives $f'(x)=0$ only at\n$x=2/\\lambda$, where $f$ attains its global maximum\n$f(2/\\lambda)=\\lambda^{2}$. \\hfill$\\square$\n\n\\smallskip\n\\textit{Proof of (R) by induction on $k$.}\n\n\\emph{Base case $k=1$.}\\;\nThe inequality reads\n\\[\n\\frac{(1+1)^{2}}{A_{1}}=\\frac{4}{a_{1}}=4\\sum_{n=1}^{1}\\frac{1}{a_{n}},\n\\]\nso (R) holds with equality when $k=1$.\n\n\\emph{Induction step.}\\;\nAssume (R) holds for $k=p\\ge 1$, that is\n\\[\n\\frac{(p+1)^{2}}{A_{p}}+\\sum_{n=1}^{p-1}\\frac{2n+1}{A_{n}}\n        \\le 4\\sum_{n=1}^{p}\\frac{1}{a_{n}}.   \\tag{I_p}\n\\]\nPut $\\lambda=p$ and $x=a_{p+1}/A_{p}$ in the lemma; after dividing the\ninequality by $A_{p}$ we obtain\n\\[\n\\frac{(p+2)^{2}}{A_{p+1}}\n      \\le\\frac{4}{a_{p+1}}+\\frac{p^{2}}{A_{p}}.  \\tag{1}\n\\]\nAdding (1) to $(I_p)$ and noting that\n\\[\n\\frac{(p+1)^{2}}{A_{p}}-\\frac{p^{2}}{A_{p}}=\\frac{2p+1}{A_{p}},\n\\]\nwe get\n\\[\n\\frac{(p+2)^{2}}{A_{p+1}}+\\sum_{n=1}^{p}\\frac{2n+1}{A_{n}}\n        \\le 4\\sum_{n=1}^{p+1}\\frac{1}{a_{n}},\n\\]\nwhich is (R) for $k=p+1$. Thus (R) holds for all $k\\ge 1$.\n\n\\smallskip\n\\textit{Passage $k\\to\\infty$ – proof of $(R_\\infty)$.}\n\n\\emph{Case $B_{\\infty}=+\\infty$.}\\;\nThen the right–hand side of (R) tends to $+\\infty$, and the limit\ninequality $\\,\\infty\\le\\infty$ is trivially true. Hence $(R_\\infty)$\nholds.\n\n\\emph{Case $B_{\\infty}<\\infty$.}\\;\nFix $k\\ge 2$ and put  \n\\[\nm:=\\Bigl\\lceil\\frac{k}{2}\\Bigr\\rceil,\\qquad \nS_{k}:=\\sum_{n=m}^{k}\\frac{1}{a_{n}}\\;=\\;B_{k}-B_{m-1}.\n\\]\nBecause $B_{\\infty}<\\infty$ we have $S_{k}\\xrightarrow[k\\to\\infty]{}0$.\nUsing Cauchy–Schwarz on the $k-m+1\\ge k/2$ terms $a_{m},\\dots,a_{k}$ we\nobtain\n\\[\n\\Bigl(\\sum_{n=m}^{k}a_{n}\\Bigr)\\Bigl(\\sum_{n=m}^{k}\\frac{1}{a_{n}}\\Bigr)\n      \\ge (k-m+1)^{2}\\ge \\frac{k^{2}}{4},\n\\]\nhence\n\\[\nA_{k}\\;\\ge\\;\\sum_{n=m}^{k}a_{n}\n      \\ge \\frac{k^{2}}{4S_{k}}.\\tag{2}\n\\]\nConsequently\n\\[\n0\\le\\frac{(k+1)^{2}}{A_{k}}\n      \\le \\frac{4(k+1)^{2}S_{k}}{k^{2}}\n      \\le 16\\,S_{k}\n      \\xrightarrow[k\\to\\infty]{}0.\\tag{3}\n\\]\nLet $k\\to\\infty$ in (R).  By monotone convergence the series on both\nsides of (R) converge to the corresponding infinite sums, while the\nfirst term on the left tends to $0$ by (3).  This yields $(R_\\infty)$.\n\n\\smallskip\n\\textit{Sharpness of the constant $4$.}\\;\nTake $a_{n}=n$; then $A_{n}=n(n+1)/2$, and\n\\[\n\\frac{2n+1}{A_{n}}\n   =\\frac{2(2n+1)}{n(n+1)}\n   =\\frac{2}{n}+\\frac{2}{n+1}.   \\tag{4}\n\\]\nHence\n\\[\n\\sum_{n=1}^{N}\\frac{2n+1}{A_{n}}\n     =2\\sum_{n=1}^{N}\\Bigl(\\frac{1}{n}+\\frac{1}{n+1}\\Bigr)\n     =4H_{N}-2+\\frac{2}{N+1},   \\tag{5}\n\\]\nwhere $H_{m}:=\\sum_{j=1}^{m}1/j$.  Because\n$\\sum_{n=1}^{N}1/a_{n}=H_{N}$, we have\n\\[\n\\frac{\\displaystyle\\sum_{n=1}^{N}\\dfrac{2n+1}{A_{n}}}\n     {\\displaystyle\\sum_{n=1}^{N}\\dfrac{1}{a_{n}}}\n     =4-\\frac{2-\\dfrac{2}{N+1}}{H_{N}}\n       \\xrightarrow[N\\to\\infty]{}4. \\tag{6}\n\\]\nTherefore no constant smaller than $4$ can satisfy (R) (hence\n$(R_\\infty)$) for all sequences, so $4$ is optimal. \\hfill$\\square$\n\n%------------------------------------------------------------\n\\textbf{Part (a) – Knopp’s inequality (K) and optimality of $2$}\n%------------------------------------------------------------\nRewrite $(R_\\infty)$ as  \n\\[\n\\sum_{n=1}^{\\infty}\\frac{2n}{A_{n}}\n      +\\sum_{n=1}^{\\infty}\\frac{1}{A_{n}}\n      \\le 4\\sum_{n=1}^{\\infty}\\frac{1}{a_{n}}. \\tag{7}\n\\]\nDrop the non–negative series $\\sum_{n\\ge 1}1/A_{n}$ and divide by $2$ to\nobtain (K).\n\nFor optimality take again $a_{n}=n$; then $A_{n}=n(n+1)/2$ and\n$n/A_{n}=2/(n+1)$.  Therefore\n\\[\n\\frac{\\displaystyle\\sum_{n=1}^{N}\\dfrac{n}{A_{n}}}\n     {\\displaystyle\\sum_{n=1}^{N}\\dfrac{1}{a_{n}}}\n     =\\frac{2(H_{N+1}-1)}{H_{N}}\n       \\xrightarrow[N\\to\\infty]{}2,\n\\]\nso the constant $2$ in (K) is best possible. \\hfill$\\square$\n\n%------------------------------------------------------------\n\\textbf{Part (b) – Strictness of (K) when $B_{\\infty}<\\infty$}\n%------------------------------------------------------------\nAssume $B_{\\infty}<\\infty$.  \nFrom (7) we isolate\n\\[\n\\sum_{n=1}^{\\infty}\\frac{n}{A_{n}}\n      \\le 2\\sum_{n=1}^{\\infty}\\frac{1}{a_{n}}\n        -\\frac{1}{2}\\sum_{n=1}^{\\infty}\\frac{1}{A_{n}}.\\tag{8}\n\\]\nBecause $A_{n}\\ge a_{n}$, we have $1/A_{n}\\le 1/a_{n}$; therefore\n$\\sum_{n\\ge 1}1/A_{n}$ converges, is strictly positive and makes the\nsecond term on the right of (8) \\emph{strictly} negative.  Hence\n\\[\n\\sum_{n=1}^{\\infty}\\frac{n}{A_{n}}\n      < 2\\sum_{n=1}^{\\infty}\\frac{1}{a_{n}},\n\\]\nso equality in (K) is impossible when $B_{\\infty}<\\infty$.\n\nIf $B_{\\infty}=+\\infty$, the right–hand side of (K) is $+\\infty$, so\nequality cannot occur either.  Thus no non–trivial strictly positive\nsequence attains equality in (K). \\hfill$\\square$\n\n\\vspace{1ex}",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.548842",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher dimensional setting – instead of one sequence we now work with an arbitrary d–dimensional array.  This multiplies the complexity of the combinatorics and forces the use of dyadic blocks in ℕᵈ.\n\n2. Additional constraints – the proof must control, simultaneously, all 2^{d} possible “halves’’ of every rectangle to obtain a uniform lower bound (Step 2).   \n\n3. Sophisticated structures – the argument employs  \n • multi–index volume estimates,  \n • a dyadic decomposition in every coordinate, and  \n • a subtle double–counting of how often each ordered entry b_s can appear.  \n\n4. Deeper theory – for sharpness (part c)) asymptotic analysis of double sums and careful estimation of rectangular partial sums are required.\n\n5. Multiple interacting concepts – combinatorial counting, analytic bounding (AM–GM), dyadic analysis and asymptotics all have to be orchestrated.  None of these appears in the one–dimensional original; together they raise the problem well beyond simple pattern-matching techniques."
      }
    },
    "original_kernel_variant": {
      "question": "Let $\\bigl(a_{n}\\bigr)_{n\\ge 1}$ be an arbitrary sequence of strictly positive real numbers and put  \n\\[\nA_{n}=a_{1}+a_{2}+\\dots +a_{n}\\qquad(n\\ge 1).\n\\]\n\na) (Knopp’s inequality – optimal form)  \nProve that  \n\\[\n\\boxed{\\;\n   \\displaystyle \n   \\sum_{n=1}^{\\infty}\\frac{n}{A_{n}}\\le 2\\sum_{n=1}^{\\infty}\\frac{1}{a_{n}}\n  \\;} \\tag{K}\n\\]\nand show that the numerical constant $2$ is best possible.\n\nb) (Impossibility of equality)  \nShow that if $\\displaystyle\\sum_{n=1}^{\\infty}1/a_{n}<\\infty$, then the\ninequality in (K) is always strict.  \n(When $\\sum 1/a_{n}=+\\infty$ the right–hand side of (K) is infinite,\nso equality is meaningless.)\n\nc) (Redheffer refinement – sharp finite and infinite versions)  \nFor every positive integer $k$ prove  \n\\[\n\\boxed{\\;\n   \\frac{(k+1)^{2}}{A_{k}}+\\sum_{n=1}^{k-1}\\frac{2n+1}{A_{n}}\n               \\le 4\\sum_{n=1}^{k}\\frac{1}{a_{n}}\n  \\;} \\tag{R}\n\\]\nand deduce the infinite counterpart  \n\\[\n\\boxed{\\;\n    \\sum_{n=1}^{\\infty}\\frac{2n+1}{A_{n}}\n           \\le 4\\sum_{n=1}^{\\infty}\\frac{1}{a_{n}}\n   \\;} \\tag{R_\\infty}\n\\]\nFinally, prove that $4$ is the best possible constant in both (R) and $(R_\\infty)$.\n\n\\vspace{1ex}",
      "solution": "Throughout write  \n\\[\ns_{n}:=\\frac{n}{A_{n}},\\qquad   \n\\Delta_{n}:=\\frac{1}{a_{n}},\\qquad\nB_{n}:=\\sum_{j=1}^{n}\\Delta_{j}\\ (n\\ge 1),\\qquad \nB_{\\infty}:=\\sum_{j\\ge 1}\\Delta_{j}.\n\\]\n\n%------------------------------------------------------------\n\\textbf{Part (c) – Redheffer’s refinement with the sharp constant $4$}\n%------------------------------------------------------------\n(The argument is self–contained and will later be used to settle parts\n(a) and (b).)\n\n\\textit{Lemma (Redheffer).}  \nFor every $x>0$ and $\\lambda>0$ one has  \n\\[\n\\frac{(\\lambda+2)^{2}}{1+x}\\le\\frac{4}{x}+\\lambda^{2},\n\\qquad\\text{with equality iff }x=\\frac{2}{\\lambda}.\n\\]\n\n\\textit{Proof.}  \nPut $f(x)=\\dfrac{(\\lambda+2)^{2}}{1+x}-\\dfrac{4}{x}$.  \nA routine derivative computation gives $f'(x)=0$ only at\n$x=2/\\lambda$, where $f$ attains its global maximum\n$f(2/\\lambda)=\\lambda^{2}$. \\hfill$\\square$\n\n\\smallskip\n\\textit{Proof of (R) by induction on $k$.}\n\n\\emph{Base case $k=1$.}\\;\nThe inequality reads\n\\[\n\\frac{(1+1)^{2}}{A_{1}}=\\frac{4}{a_{1}}=4\\sum_{n=1}^{1}\\frac{1}{a_{n}},\n\\]\nso (R) holds with equality when $k=1$.\n\n\\emph{Induction step.}\\;\nAssume (R) holds for $k=p\\ge 1$, that is\n\\[\n\\frac{(p+1)^{2}}{A_{p}}+\\sum_{n=1}^{p-1}\\frac{2n+1}{A_{n}}\n        \\le 4\\sum_{n=1}^{p}\\frac{1}{a_{n}}.   \\tag{I_p}\n\\]\nPut $\\lambda=p$ and $x=a_{p+1}/A_{p}$ in the lemma; after dividing the\ninequality by $A_{p}$ we obtain\n\\[\n\\frac{(p+2)^{2}}{A_{p+1}}\n      \\le\\frac{4}{a_{p+1}}+\\frac{p^{2}}{A_{p}}.  \\tag{1}\n\\]\nAdding (1) to $(I_p)$ and noting that\n\\[\n\\frac{(p+1)^{2}}{A_{p}}-\\frac{p^{2}}{A_{p}}=\\frac{2p+1}{A_{p}},\n\\]\nwe get\n\\[\n\\frac{(p+2)^{2}}{A_{p+1}}+\\sum_{n=1}^{p}\\frac{2n+1}{A_{n}}\n        \\le 4\\sum_{n=1}^{p+1}\\frac{1}{a_{n}},\n\\]\nwhich is (R) for $k=p+1$. Thus (R) holds for all $k\\ge 1$.\n\n\\smallskip\n\\textit{Passage $k\\to\\infty$ – proof of $(R_\\infty)$.}\n\n\\emph{Case $B_{\\infty}=+\\infty$.}\\;\nThen the right–hand side of (R) tends to $+\\infty$, and the limit\ninequality $\\,\\infty\\le\\infty$ is trivially true. Hence $(R_\\infty)$\nholds.\n\n\\emph{Case $B_{\\infty}<\\infty$.}\\;\nFix $k\\ge 2$ and put  \n\\[\nm:=\\Bigl\\lceil\\frac{k}{2}\\Bigr\\rceil,\\qquad \nS_{k}:=\\sum_{n=m}^{k}\\frac{1}{a_{n}}\\;=\\;B_{k}-B_{m-1}.\n\\]\nBecause $B_{\\infty}<\\infty$ we have $S_{k}\\xrightarrow[k\\to\\infty]{}0$.\nUsing Cauchy–Schwarz on the $k-m+1\\ge k/2$ terms $a_{m},\\dots,a_{k}$ we\nobtain\n\\[\n\\Bigl(\\sum_{n=m}^{k}a_{n}\\Bigr)\\Bigl(\\sum_{n=m}^{k}\\frac{1}{a_{n}}\\Bigr)\n      \\ge (k-m+1)^{2}\\ge \\frac{k^{2}}{4},\n\\]\nhence\n\\[\nA_{k}\\;\\ge\\;\\sum_{n=m}^{k}a_{n}\n      \\ge \\frac{k^{2}}{4S_{k}}.\\tag{2}\n\\]\nConsequently\n\\[\n0\\le\\frac{(k+1)^{2}}{A_{k}}\n      \\le \\frac{4(k+1)^{2}S_{k}}{k^{2}}\n      \\le 16\\,S_{k}\n      \\xrightarrow[k\\to\\infty]{}0.\\tag{3}\n\\]\nLet $k\\to\\infty$ in (R).  By monotone convergence the series on both\nsides of (R) converge to the corresponding infinite sums, while the\nfirst term on the left tends to $0$ by (3).  This yields $(R_\\infty)$.\n\n\\smallskip\n\\textit{Sharpness of the constant $4$.}\\;\nTake $a_{n}=n$; then $A_{n}=n(n+1)/2$, and\n\\[\n\\frac{2n+1}{A_{n}}\n   =\\frac{2(2n+1)}{n(n+1)}\n   =\\frac{2}{n}+\\frac{2}{n+1}.   \\tag{4}\n\\]\nHence\n\\[\n\\sum_{n=1}^{N}\\frac{2n+1}{A_{n}}\n     =2\\sum_{n=1}^{N}\\Bigl(\\frac{1}{n}+\\frac{1}{n+1}\\Bigr)\n     =4H_{N}-2+\\frac{2}{N+1},   \\tag{5}\n\\]\nwhere $H_{m}:=\\sum_{j=1}^{m}1/j$.  Because\n$\\sum_{n=1}^{N}1/a_{n}=H_{N}$, we have\n\\[\n\\frac{\\displaystyle\\sum_{n=1}^{N}\\dfrac{2n+1}{A_{n}}}\n     {\\displaystyle\\sum_{n=1}^{N}\\dfrac{1}{a_{n}}}\n     =4-\\frac{2-\\dfrac{2}{N+1}}{H_{N}}\n       \\xrightarrow[N\\to\\infty]{}4. \\tag{6}\n\\]\nTherefore no constant smaller than $4$ can satisfy (R) (hence\n$(R_\\infty)$) for all sequences, so $4$ is optimal. \\hfill$\\square$\n\n%------------------------------------------------------------\n\\textbf{Part (a) – Knopp’s inequality (K) and optimality of $2$}\n%------------------------------------------------------------\nRewrite $(R_\\infty)$ as  \n\\[\n\\sum_{n=1}^{\\infty}\\frac{2n}{A_{n}}\n      +\\sum_{n=1}^{\\infty}\\frac{1}{A_{n}}\n      \\le 4\\sum_{n=1}^{\\infty}\\frac{1}{a_{n}}. \\tag{7}\n\\]\nDrop the non–negative series $\\sum_{n\\ge 1}1/A_{n}$ and divide by $2$ to\nobtain (K).\n\nFor optimality take again $a_{n}=n$; then $A_{n}=n(n+1)/2$ and\n$n/A_{n}=2/(n+1)$.  Therefore\n\\[\n\\frac{\\displaystyle\\sum_{n=1}^{N}\\dfrac{n}{A_{n}}}\n     {\\displaystyle\\sum_{n=1}^{N}\\dfrac{1}{a_{n}}}\n     =\\frac{2(H_{N+1}-1)}{H_{N}}\n       \\xrightarrow[N\\to\\infty]{}2,\n\\]\nso the constant $2$ in (K) is best possible. \\hfill$\\square$\n\n%------------------------------------------------------------\n\\textbf{Part (b) – Strictness of (K) when $B_{\\infty}<\\infty$}\n%------------------------------------------------------------\nAssume $B_{\\infty}<\\infty$.  \nFrom (7) we isolate\n\\[\n\\sum_{n=1}^{\\infty}\\frac{n}{A_{n}}\n      \\le 2\\sum_{n=1}^{\\infty}\\frac{1}{a_{n}}\n        -\\frac{1}{2}\\sum_{n=1}^{\\infty}\\frac{1}{A_{n}}.\\tag{8}\n\\]\nBecause $A_{n}\\ge a_{n}$, we have $1/A_{n}\\le 1/a_{n}$; therefore\n$\\sum_{n\\ge 1}1/A_{n}$ converges, is strictly positive and makes the\nsecond term on the right of (8) \\emph{strictly} negative.  Hence\n\\[\n\\sum_{n=1}^{\\infty}\\frac{n}{A_{n}}\n      < 2\\sum_{n=1}^{\\infty}\\frac{1}{a_{n}},\n\\]\nso equality in (K) is impossible when $B_{\\infty}<\\infty$.\n\nIf $B_{\\infty}=+\\infty$, the right–hand side of (K) is $+\\infty$, so\nequality cannot occur either.  Thus no non–trivial strictly positive\nsequence attains equality in (K). \\hfill$\\square$\n\n\\vspace{1ex}",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.454535",
        "was_fixed": false,
        "difficulty_analysis": "1. Higher dimensional setting – instead of one sequence we now work with an arbitrary d–dimensional array.  This multiplies the complexity of the combinatorics and forces the use of dyadic blocks in ℕᵈ.\n\n2. Additional constraints – the proof must control, simultaneously, all 2^{d} possible “halves’’ of every rectangle to obtain a uniform lower bound (Step 2).   \n\n3. Sophisticated structures – the argument employs  \n • multi–index volume estimates,  \n • a dyadic decomposition in every coordinate, and  \n • a subtle double–counting of how often each ordered entry b_s can appear.  \n\n4. Deeper theory – for sharpness (part c)) asymptotic analysis of double sums and careful estimation of rectangular partial sums are required.\n\n5. Multiple interacting concepts – combinatorial counting, analytic bounding (AM–GM), dyadic analysis and asymptotics all have to be orchestrated.  None of these appears in the one–dimensional original; together they raise the problem well beyond simple pattern-matching techniques."
      }
    }
  },
  "checked": true,
  "problem_type": "proof",
  "iteratively_fixed": true
}