{
  "index": "1974-B-5",
  "type": "ANA",
  "tag": [
    "ANA",
    "ALG"
  ],
  "difficulty": "",
  "question": "B-5. Show that \\( 1+(n / 1!)+\\left(n^{2} / 2!\\right)+\\cdots+\\left(n^{n} / n!\\right)>e^{n} / 2 \\) for every integer \\( n \\geqq 0 \\).\nRemarks You may assume as known Taylor's remainder formula:\n\\[\ne^{x}-\\sum_{k=0}^{n} \\frac{x^{k}}{k!}=\\frac{1}{n!} \\int_{0}^{x}(x-t)^{n} e^{\\prime} d t\n\\]\nas well as the fact that\n\\[\nn!=\\int_{0}^{-} t^{n} e^{-t} d t\n\\]",
  "solution": "B-5.\nWe want to show that\n\\[\n\\sum_{k=0}^{n} \\frac{n^{k}}{k!}=e^{n}-\\frac{1}{n!} \\int_{0}^{n}(n-t)^{n} e^{t} d t>\\frac{e^{n}}{2}\n\\]\nor, equivalently, that\n\\[\n\\begin{array}{l}\nn!>2 e^{-n} \\int_{0}^{n}(n-t)^{n} e^{t} d t \\\\\n\\int_{0}^{\\infty} t^{n} e^{-t} d t>2 e^{-n} \\int_{0}^{n}(n-t)^{n} e^{t} d t .\n\\end{array}\n\\]\n\nLetting \\( u=n-t \\), this can be transformed into\n\\[\n\\int_{0}^{\\infty} t^{n} e^{-t} d t>2 \\int_{0}^{n} u^{n} e^{-u} d u\n\\]\nwhich is equivalent to\n\\[\n\\int_{n}^{x} u^{n} e^{-u} d u>\\int_{0}^{n} u^{n} e^{-u} d u .\n\\]\n\nLet \\( f(u)=u^{n} e^{-u} \\). Then it suffices to show that\n\\[\nf(n+h) \\geqq f(n-h) \\quad \\text { for } \\quad 0 \\leqq h \\leqq n .\n\\]\n\nThis is equivalent to\n\\[\n\\begin{array}{l}\n(n+h)^{n} e^{-n} \\geqq(n-h)^{n} e^{n} . \\\\\nn \\ln (n+h)-h \\geqq n \\ln (n-h)+h .\n\\end{array}\n\\]\n\nLet \\( g(h)=n \\ln (n+h)-n \\ln (n-h)-2 h \\). Then \\( g(0)=0 \\) and\n\\[\n\\frac{d g}{d h}=\\frac{n}{n+h}+\\frac{n}{n-h}-2=\\frac{2 n^{2}}{n^{2}-h^{2}}-2>0\n\\]\nfor \\( 0<h<n \\). Hence \\( g(h)>0 \\) for \\( 0<h<n \\). The desired result follows.",
  "vars": [
    "x",
    "k",
    "t",
    "u",
    "h",
    "f",
    "g"
  ],
  "params": [
    "n"
  ],
  "sci_consts": [
    "e"
  ],
  "variants": {
    "descriptive_long": {
      "map": {
        "x": "variablex",
        "k": "indexvar",
        "t": "timevar",
        "u": "shiftvar",
        "h": "offsetvar",
        "f": "densityfn",
        "g": "growthfn",
        "n": "fixedint"
      },
      "question": "B-5. Show that \\( 1+(fixedint / 1!)+\\left(fixedint^{2} / 2!\\right)+\\cdots+\\left(fixedint^{fixedint} / fixedint!\\right)>e^{fixedint} / 2 \\) for every integer \\( fixedint \\geqq 0 \\).\nRemarks You may assume as known Taylor's remainder formula:\n\\[\n e^{variablex}-\\sum_{indexvar=0}^{fixedint} \\frac{variablex^{indexvar}}{indexvar!}=\\frac{1}{fixedint!} \\int_{0}^{variablex}(variablex-timevar)^{fixedint} e^{\\prime} d timevar\n\\]\nas well as the fact that\n\\[\n fixedint!=\\int_{0}^{-} timevar^{fixedint} e^{-timevar} d timevar\n\\]",
      "solution": "B-5.\nWe want to show that\n\\[\n\\sum_{indexvar=0}^{fixedint} \\frac{fixedint^{indexvar}}{indexvar!}=e^{fixedint}-\\frac{1}{fixedint!} \\int_{0}^{fixedint}(fixedint-timevar)^{fixedint} e^{timevar} d timevar>\\frac{e^{fixedint}}{2}\n\\]\nor, equivalently, that\n\\[\n\\begin{array}{l}\nfixedint!>2 e^{-fixedint} \\int_{0}^{fixedint}(fixedint-timevar)^{fixedint} e^{timevar} d timevar \\\\\n\\int_{0}^{\\infty} timevar^{fixedint} e^{-timevar} d timevar>2 e^{-fixedint} \\int_{0}^{fixedint}(fixedint-timevar)^{fixedint} e^{timevar} d timevar .\n\\end{array}\n\\]\n\nLetting \\( shiftvar=fixedint-timevar \\), this can be transformed into\n\\[\n\\int_{0}^{\\infty} timevar^{fixedint} e^{-timevar} d timevar>2 \\int_{0}^{fixedint} shiftvar^{fixedint} e^{-shiftvar} d shiftvar\n\\]\nwhich is equivalent to\n\\[\n\\int_{fixedint}^{variablex} shiftvar^{fixedint} e^{-shiftvar} d shiftvar>\\int_{0}^{fixedint} shiftvar^{fixedint} e^{-shiftvar} d shiftvar .\n\\]\n\nLet \\( densityfn(shiftvar)=shiftvar^{fixedint} e^{-shiftvar} \\). Then it suffices to show that\n\\[\n densityfn(fixedint+offsetvar) \\geqq densityfn(fixedint-offsetvar) \\quad \\text { for } \\quad 0 \\leqq offsetvar \\leqq fixedint .\n\\]\n\nThis is equivalent to\n\\[\n\\begin{array}{l}\n(fixedint+offsetvar)^{fixedint} e^{-fixedint} \\geqq(fixedint-offsetvar)^{fixedint} e^{fixedint} . \\\\\nfixedint \\ln (fixedint+offsetvar)-offsetvar \\geqq fixedint \\ln (fixedint-offsetvar)+offsetvar .\n\\end{array}\n\\]\n\nLet \\( growthfn(offsetvar)=fixedint \\ln (fixedint+offsetvar)-fixedint \\ln (fixedint-offsetvar)-2 offsetvar \\). Then \\( growthfn(0)=0 \\) and\n\\[\n\\frac{d growthfn}{d offsetvar}=\\frac{fixedint}{fixedint+offsetvar}+\\frac{fixedint}{fixedint-offsetvar}-2=\\frac{2 fixedint^{2}}{fixedint^{2}-offsetvar^{2}}-2>0\n\\]\nfor \\( 0<offsetvar<fixedint \\). Hence \\( growthfn(offsetvar)>0 \\) for \\( 0<offsetvar<fixedint \\). The desired result follows."
    },
    "descriptive_long_confusing": {
      "map": {
        "x": "paperclip",
        "k": "turntable",
        "t": "goldfish",
        "u": "horsehair",
        "h": "toothpick",
        "f": "chandelier",
        "g": "sandstorm",
        "n": "blueberry"
      },
      "question": "B-5. Show that \\( 1+(blueberry / 1!)+\\left(blueberry^{2} / 2!\\right)+\\cdots+\\left(blueberry^{blueberry} / blueberry!\\right)>e^{blueberry} / 2 \\) for every integer \\( blueberry \\geqq 0 \\).\nRemarks You may assume as known Taylor's remainder formula:\n\\[\ne^{paperclip}-\\sum_{turntable=0}^{blueberry} \\frac{paperclip^{turntable}}{turntable!}=\\frac{1}{blueberry!} \\int_{0}^{paperclip}(paperclip-goldfish)^{blueberry} e^{\\prime} d goldfish\n\\]\nas well as the fact that\n\\[\nblueberry!=\\int_{0}^{-} goldfish^{blueberry} e^{-goldfish} d goldfish\n\\]",
      "solution": "B-5.\nWe want to show that\n\\[\n\\sum_{turntable=0}^{blueberry} \\frac{blueberry^{turntable}}{turntable!}=e^{blueberry}-\\frac{1}{blueberry!} \\int_{0}^{blueberry}(blueberry-goldfish)^{blueberry} e^{goldfish} d goldfish>\\frac{e^{blueberry}}{2}\n\\]\nor, equivalently, that\n\\[\n\\begin{array}{l}\nblueberry!>2 e^{-blueberry} \\int_{0}^{blueberry}(blueberry-goldfish)^{blueberry} e^{goldfish} d goldfish \\\\\n\\int_{0}^{\\infty} goldfish^{blueberry} e^{-goldfish} d goldfish>2 e^{-blueberry} \\int_{0}^{blueberry}(blueberry-goldfish)^{blueberry} e^{goldfish} d goldfish .\n\\end{array}\n\\]\n\nLetting \\( horsehair=blueberry-goldfish \\), this can be transformed into\n\\[\n\\int_{0}^{\\infty} goldfish^{blueberry} e^{-goldfish} d goldfish>2 \\int_{0}^{blueberry} horsehair^{blueberry} e^{-horsehair} d horsehair\n\\]\nwhich is equivalent to\n\\[\n\\int_{blueberry}^{paperclip} horsehair^{blueberry} e^{-horsehair} d horsehair>\\int_{0}^{blueberry} horsehair^{blueberry} e^{-horsehair} d horsehair .\n\\]\n\nLet \\( chandelier(horsehair)=horsehair^{blueberry} e^{-horsehair} \\). Then it suffices to show that\n\\[\nchandelier(blueberry+toothpick) \\geqq chandelier(blueberry-toothpick) \\quad \\text { for } \\quad 0 \\leqq toothpick \\leqq blueberry .\n\\]\n\nThis is equivalent to\n\\[\n\\begin{array}{l}\n(blueberry+toothpick)^{blueberry} e^{-blueberry} \\geqq(blueberry-toothpick)^{blueberry} e^{blueberry} . \\\\\nblueberry \\ln (blueberry+toothpick)-toothpick \\geqq blueberry \\ln (blueberry-toothpick)+toothpick .\n\\end{array}\n\\]\n\nLet \\( sandstorm(toothpick)=blueberry \\ln (blueberry+toothpick)-blueberry \\ln (blueberry-toothpick)-2 toothpick \\). Then \\( sandstorm(0)=0 \\) and\n\\[\n\\frac{d sandstorm}{d toothpick}=\\frac{blueberry}{blueberry+toothpick}+\\frac{blueberry}{blueberry-toothpick}-2=\\frac{2 blueberry^{2}}{blueberry^{2}-toothpick^{2}}-2>0\n\\]\nfor \\( 0<toothpick<blueberry \\). Hence \\( sandstorm(toothpick)>0 \\) for \\( 0<toothpick<blueberry \\). The desired result follows."
    },
    "descriptive_long_misleading": {
      "map": {
        "x": "constantvalue",
        "k": "totalcount",
        "t": "staticpoint",
        "u": "baselinepoint",
        "h": "unchanged",
        "f": "staticnumber",
        "g": "constantstate",
        "n": "zerovalue"
      },
      "question": "B-5. Show that \\( 1+(\\zerovalue / 1!)+\\left(\\zerovalue^{2} / 2!\\right)+\\cdots+\\left(\\zerovalue^{\\zerovalue} / \\zerovalue!\\right)>e^{\\zerovalue} / 2 \\) for every integer \\( \\zerovalue \\geqq 0 \\).\nRemarks You may assume as known Taylor's remainder formula:\n\\[\ne^{constantvalue}-\\sum_{totalcount=0}^{\\zerovalue} \\frac{constantvalue^{totalcount}}{totalcount!}=\\frac{1}{\\zerovalue!} \\int_{0}^{constantvalue}(constantvalue-staticpoint)^{\\zerovalue} e^{\\prime} d staticpoint\n\\]\nas well as the fact that\n\\[\n\\zerovalue!=\\int_{0}^{-} staticpoint^{\\zerovalue} e^{-staticpoint} d staticpoint\n\\]",
      "solution": "B-5.\nWe want to show that\n\\[\n\\sum_{totalcount=0}^{\\zerovalue} \\frac{\\zerovalue^{totalcount}}{totalcount!}=e^{\\zerovalue}-\\frac{1}{\\zerovalue!} \\int_{0}^{\\zerovalue}(\\zerovalue-staticpoint)^{\\zerovalue} e^{staticpoint} d staticpoint>\\frac{e^{\\zerovalue}}{2}\n\\]\nor, equivalently, that\n\\[\n\\begin{array}{l}\n\\zerovalue!>2 e^{-\\zerovalue} \\int_{0}^{\\zerovalue}(\\zerovalue-staticpoint)^{\\zerovalue} e^{staticpoint} d staticpoint \\\\\n\\int_{0}^{\\infty} staticpoint^{\\zerovalue} e^{-staticpoint} d staticpoint>2 e^{-\\zerovalue} \\int_{0}^{\\zerovalue} baselinepoint^{\\zerovalue} e^{-baselinepoint} d baselinepoint .\n\\end{array}\n\\]\n\nLetting \\( baselinepoint=\\zerovalue-staticpoint \\), this can be transformed into\n\\[\n\\int_{0}^{\\infty} staticpoint^{\\zerovalue} e^{-staticpoint} d staticpoint>2 \\int_{0}^{\\zerovalue} baselinepoint^{\\zerovalue} e^{-baselinepoint} d baselinepoint\n\\]\nwhich is equivalent to\n\\[\n\\int_{\\zerovalue}^{constantvalue} baselinepoint^{\\zerovalue} e^{-baselinepoint} d baselinepoint>\\int_{0}^{\\zerovalue} baselinepoint^{\\zerovalue} e^{-baselinepoint} d baselinepoint .\n\\]\n\nLet \\( staticnumber(baselinepoint)=baselinepoint^{\\zerovalue} e^{-baselinepoint} \\). Then it suffices to show that\n\\[\nstaticnumber(\\zerovalue+unchanged) \\geqq staticnumber(\\zerovalue-unchanged) \\quad \\text { for } \\quad 0 \\leqq unchanged \\leqq \\zerovalue .\n\\]\n\nThis is equivalent to\n\\[\n\\begin{array}{l}\n(\\zerovalue+unchanged)^{\\zerovalue} e^{-\\zerovalue} \\geqq(\\zerovalue-unchanged)^{\\zerovalue} e^{\\zerovalue} . \\\\\n\\zerovalue \\ln (\\zerovalue+unchanged)-unchanged \\geqq \\zerovalue \\ln (\\zerovalue-unchanged)+unchanged .\n\\end{array}\n\\]\n\nLet \\( constantstate(unchanged)=\\zerovalue \\ln (\\zerovalue+unchanged)-\\zerovalue \\ln (\\zerovalue-unchanged)-2 unchanged \\). Then \\( constantstate(0)=0 \\) and\n\\[\n\\frac{d constantstate}{d unchanged}=\\frac{\\zerovalue}{\\zerovalue+unchanged}+\\frac{\\zerovalue}{\\zerovalue-unchanged}-2=\\frac{2 \\zerovalue^{2}}{\\zerovalue^{2}-unchanged^{2}}-2>0\n\\]\nfor \\( 0<unchanged<\\zerovalue \\). Hence \\( constantstate(unchanged)>0 \\) for \\( 0<unchanged<\\zerovalue \\). The desired result follows."
    },
    "garbled_string": {
      "map": {
        "x": "qzxwvtnp",
        "k": "lmnoprqs",
        "t": "hjgrksla",
        "u": "vcbnerpm",
        "h": "sdfkqwer",
        "f": "zxcvbnml",
        "g": "pokiuytr",
        "n": "asdfghjk"
      },
      "question": "B-5. Show that \\( 1+(asdfghjk / 1!)+\\left(asdfghjk^{2} / 2!\\right)+\\cdots+\\left(asdfghjk^{asdfghjk} / asdfghjk!\\right)>e^{asdfghjk} / 2 \\) for every integer \\( asdfghjk \\geqq 0 \\).\nRemarks You may assume as known Taylor's remainder formula:\n\\[\ne^{qzxwvtnp}-\\sum_{lmnoprqs=0}^{asdfghjk} \\frac{qzxwvtnp^{lmnoprqs}}{lmnoprqs!}=\\frac{1}{asdfghjk!} \\int_{0}^{qzxwvtnp}(qzxwvtnp-hjgrksla)^{asdfghjk} e^{\\prime} d hjgrksla\n\\]\nas well as the fact that\n\\[\nasdfghjk!=\\int_{0}^{-} hjgrksla^{asdfghjk} e^{-hjgrksla} d hjgrksla\n\\]\n",
      "solution": "B-5.\nWe want to show that\n\\[\n\\sum_{lmnoprqs=0}^{asdfghjk} \\frac{asdfghjk^{lmnoprqs}}{lmnoprqs!}=e^{asdfghjk}-\\frac{1}{asdfghjk!} \\int_{0}^{asdfghjk}(asdfghjk-hjgrksla)^{asdfghjk} e^{hjgrksla} d hjgrksla>\\frac{e^{asdfghjk}}{2}\n\\]\nor, equivalently, that\n\\[\n\\begin{array}{l}\nasdfghjk!>2 e^{-asdfghjk} \\int_{0}^{asdfghjk}(asdfghjk-hjgrksla)^{asdfghjk} e^{hjgrksla} d hjgrksla \\\\\n\\int_{0}^{\\infty} hjgrksla^{asdfghjk} e^{-hjgrksla} d hjgrksla>2 e^{-asdfghjk} \\int_{0}^{asdfghjk}(asdfghjk-hjgrksla)^{asdfghjk} e^{hjgrksla} d hjgrksla .\n\\end{array}\n\\]\n\nLetting \\( vcbnerpm=asdfghjk-hjgrksla \\), this can be transformed into\n\\[\n\\int_{0}^{\\infty} hjgrksla^{asdfghjk} e^{-hjgrksla} d hjgrksla>2 \\int_{0}^{asdfghjk} vcbnerpm^{asdfghjk} e^{-vcbnerpm} d vcbnerpm\n\\]\nwhich is equivalent to\n\\[\n\\int_{asdfghjk}^{qzxwvtnp} vcbnerpm^{asdfghjk} e^{-vcbnerpm} d vcbnerpm>\\int_{0}^{asdfghjk} vcbnerpm^{asdfghjk} e^{-vcbnerpm} d vcbnerpm .\n\\]\n\nLet \\( zxcvbnml(vcbnerpm)=vcbnerpm^{asdfghjk} e^{-vcbnerpm} \\). Then it suffices to show that\n\\[\nzxcvbnml(asdfghjk+sdfkqwer) \\geqq zxcvbnml(asdfghjk-sdfkqwer) \\quad \\text { for } \\quad 0 \\leqq sdfkqwer \\leqq asdfghjk .\n\\]\n\nThis is equivalent to\n\\[\n\\begin{array}{l}\n(asdfghjk+sdfkqwer)^{asdfghjk} e^{-asdfghjk} \\geqq(asdfghjk-sdfkqwer)^{asdfghjk} e^{asdfghjk} . \\\\\nasdfghjk \\ln (asdfghjk+sdfkqwer)-sdfkqwer \\geqq asdfghjk \\ln (asdfghjk-sdfkqwer)+sdfkqwer .\n\\end{array}\n\\]\n\nLet \\( pokiuytr(sdfkqwer)=asdfghjk \\ln (asdfghjk+sdfkqwer)-asdfghjk \\ln (asdfghjk-sdfkqwer)-2 sdfkqwer \\). Then \\( pokiuytr(0)=0 \\) and\n\\[\n\\frac{d pokiuytr}{d sdfkqwer}=\\frac{asdfghjk}{asdfghjk+sdfkqwer}+\\frac{asdfghjk}{asdfghjk-sdfkqwer}-2=\\frac{2 asdfghjk^{2}}{asdfghjk^{2}-sdfkqwer^{2}}-2>0\n\\]\nfor \\( 0<sdfkqwer<asdfghjk \\). Hence \\( pokiuytr(sdfkqwer)>0 \\) for \\( 0<sdfkqwer<asdfghjk \\). The desired result follows.\n"
    },
    "kernel_variant": {
      "question": "Let  \n\\[\nS_{n}:=\\sum_{k=0}^{\\,n}\\frac{n^{k}}{k!},\\qquad  \nP_{n}:=e^{-n}S_{n}=\\mathbb P\\!\\left\\{X_{n}\\le n\\right\\},\n\\qquad X_{n}\\sim\\text{\\rm Poisson}(n),\\qquad n\\in\\mathbb N .\n\\]\n\n(a)  Prove that the sequence $\\bigl(P_{n}\\bigr)_{n\\ge 0}$ is strictly decreasing and that  \n\\[\n\\lim_{n\\to\\infty}P_{n}=\\frac12 .\n\\]\n\n(b)  Show the quantitative two–sided bound  \n\\[\n\\boxed{\\;\n\\frac12+\\frac{1}{7\\sqrt n}\n\\;<\\;\nP_{n}\n\\;<\\;\n\\frac12+\\frac{1}{2\\sqrt n}\n\\;}\\qquad(n\\ge 3).\n\\]\n\n(c)  Deduce the sharpened estimate  \n\\[\nS_{n}>e^{n}\\!\n\\left(\n\\frac12+\\frac{1}{7\\sqrt n}\n\\right)\n\\qquad(n\\ge 3).\n\\]\n\nYou may use without proof  \n\n(i)  the Gamma–integral $\\Gamma(m+1)=\\displaystyle\\int_{0}^{\\infty}t^{m}e^{-t}\\,dt$,  \n\n(ii) Stirling’s two–sided estimate  \n\\[\n\\sqrt{2\\pi}\\,m^{\\,m+\\frac12}e^{-m}\n<\nm!\n<\n\\sqrt{2\\pi}\\,m^{\\,m+\\frac12}e^{-m}\\exp\\!\\bigl(\\tfrac{1}{12m}\\bigr)\n\\qquad(m\\ge 1),\n\\]\n\n(iii) the fact that the Poisson probability mass function is unimodal, its maximum being attained at $k=n$ (and also at $k=n-1$ when $n\\ge1$).\n\n\\bigskip",
      "solution": "\\textbf{Step 0.  Probabilistic reformulation}\n\nFor $X_{n}\\sim\\text{\\rm Poisson}(n)$,\n\\[\nP_{n}=e^{-n}\\sum_{k=0}^{n}\\frac{n^{k}}{k!}\n      =\\mathbb P\\!\\bigl\\{X_{n}\\le n\\bigr\\},\n\\qquad\nS_{n}=e^{n}P_{n}.\n\\]\n\n\\bigskip\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\textbf{Step 1.  A \\emph{correct} integral representation}\n\nDenote by  \n\\[\n\\Gamma(s,x)=\\int_{x}^{\\infty}t^{s-1}e^{-t}\\,dt\\qquad(s>0,\\;x\\ge0)\n\\]\nthe \\emph{upper} incomplete Gamma function.  \nSince $\\Gamma(n+1)=n!$, changing the dummy letter gives  \n\\[\nP_{n}\n  =e^{-n}\\sum_{k=0}^{n}\\frac{n^{k}}{k!}\n  =\\frac{\\Gamma(n+1,n)}{n!}\n  =\\frac{1}{n!}\\int_{n}^{\\infty}t^{n}e^{-t}\\,dt.\n\\tag{1}\n\\]\n\n\\bigskip\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\textbf{Part (a).  Strict decrease and limit}\n\n\\emph{(i)  Strict decrease.}\nUsing (1) for $n$ and $n+1$ we obtain\n\\[\nP_{n}-P_{n+1}\n   =\\frac{1}{n!}\\!\\int_{n}^{\\infty}\\!t^{n}e^{-t}\\,dt\n    -\\frac{1}{(n+1)!}\\!\\int_{\\,n+1}^{\\infty}\\!t^{\\,n+1}e^{-t}\\,dt\n   =\\frac{1}{(n+1)!}\\Bigl[I_{n}-n^{\\,n+1}e^{-n}\\Bigr],\n\\]\nwhere  \n\\[\nI_{n}:=\\int_{n}^{\\,n+1}t^{\\,n+1}e^{-t}\\,dt.\n\\tag{2}\n\\]\n\n\\emph{Claim.}  For every $n\\ge0$ one has $I_{n}>n^{\\,n+1}e^{-n}$.\n\n\\emph{Proof of the claim.}\nPut $t=n+s$ with $0\\le s\\le1$; then\n\\[\nI_{n}=n^{\\,n+1}e^{-n}\\!\\int_{0}^{1}\n          \\Bigl(1+\\frac{s}{n}\\Bigr)^{n+1}e^{-s}\\,ds.\n\\]\nDefine  \n\\[\nf_{n}(s):=\\Bigl(1+\\frac{s}{n}\\Bigr)^{n+1}e^{-s},\\qquad 0\\le s\\le1 .\n\\]\nA simple derivative computation gives\n\\[\nf_{n}'(s)=f_{n}(s)\\,\\frac{1-s}{n+s}\\;>\\;0\n\\qquad(0<s<1),\n\\]\nso $f_{n}$ is \\emph{strictly increasing} on $[0,1]$ and $f_{n}(0)=1$.  \nHence $f_{n}(s)>1$ for $0<s\\le1$, and\n\\[\nI_{n}=n^{\\,n+1}e^{-n}\\!\\int_{0}^{1}f_{n}(s)\\,ds\n     >n^{\\,n+1}e^{-n}\\!\\int_{0}^{1}1\\,ds\n     =n^{\\,n+1}e^{-n}.\n\\]\nThe claim is proved.\n\nBecause the front factor $1/(n+1)!$ is positive, the bracket in (2) is\npositive, and\n\\[\nP_{n}>P_{n+1}\\qquad(n\\ge0).\n\\]\nStrict decrease is now established.\n\n\\smallskip\n\\emph{(ii)  The limit.}\nStandardising $X_{n}$,\n\\[\nZ_{n}:=\\frac{X_{n}-n}{\\sqrt n},\n\\]\nthe classical Central Limit Theorem yields $Z_{n}\\Longrightarrow N(0,1)$.\nSince $P_{n}=\\mathbb P\\{Z_{n}\\le0\\}$ and the Gaussian limit is symmetric,\n\\[\n\\lim_{n\\to\\infty}P_{n}=\\Phi(0)=\\frac12.\n\\]\n\n\\bigskip\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\textbf{Part (b).  A two–sided $\\sqrt n$–law}\n\nSet\n\\[\np_{n}:=\\mathbb P\\!\\bigl\\{X_{n}=n\\bigr\\}\n      =e^{-n}\\frac{n^{n}}{n!},\\qquad\na_{n}:=\\mathbb P\\!\\bigl\\{X_{n}\\le n-1\\bigr\\},\\qquad\nc_{n}:=\\mathbb P\\!\\bigl\\{X_{n}\\ge n+1\\bigr\\},\n\\]\nso that\n\\[\nP_{n}=a_{n}+p_{n},\\qquad\n1=a_{n}+p_{n}+c_{n}.\n\\]\n\n\\medskip\n\\emph{Lemma 1 (right tail dominates the far left).}\nFor every $n\\ge1$ and $h\\ge0$,\n\\[\n\\mathbb P\\!\\bigl\\{X_{n}=n+h\\bigr\\}\\;\\ge\\;\n\\mathbb P\\!\\bigl\\{X_{n}=n-1-h\\bigr\\}.\n\\tag{3}\n\\]\n\n\\emph{Proof.}\nUsing the explicit density,\n\\[\nR_{h}:=\n\\frac{\\mathbb P\\{X_{n}=n+h\\}}{\\mathbb P\\{X_{n}=n-1-h\\}}\n  =n^{\\,2h+1}\\frac{(n-1-h)!}{(n+h)!}\n  =\\prod_{j=0}^{2h}\\frac{n}{n-h+j}.\n\\]\nPair the symmetric factors:\n\\[\n(n-h+j)\\,(n+h-j)\\le n^{2}\\qquad(0\\le j\\le h),\n\\]\nso each product of two consecutive denominators does not exceed $n^{2}$\nand therefore the whole denominator does not exceed the numerator\n$n^{\\,2h+1}$.  \nHence $R_{h}\\ge1$, proving (3).\n\nSumming (3) over $h\\ge0$ gives\n\\[\np_{n}+c_{n}\\;\\ge\\;a_{n}.\n\\tag{4}\n\\]\n\n\\smallskip\n\\emph{Lemma 2 (left side dominates the symmetric right).}\nFor every $n\\ge1$ and $h\\ge1$,\n\\[\n\\mathbb P\\!\\bigl\\{X_{n}=n-h\\bigr\\}\\;\\ge\\;\n\\mathbb P\\!\\bigl\\{X_{n}=n+h\\bigr\\}.\n\\tag{5}\n\\]\n\\emph{Proof.}\nThe Poisson mass function is strictly increasing up to $k=n$ and\nstrictly decreasing afterwards (Fact (iii)), whence (5).\\hfill$\\square$\n\nSumming (5) over $h\\ge1$ yields\n\\[\na_{n}\\;\\ge\\;c_{n}.\n\\tag{6}\n\\]\n\n\\smallskip\n\\emph{Consequences of the two lemmas.}\nFrom (4) we get $a_{n}\\le p_{n}+c_{n}$, while (6) implies $c_{n}\\le a_{n}$.\nCombining,\n\\[\na_{n}\\le\\frac12,\n\\qquad\na_{n}\\ge\\frac{1-p_{n}}{2}.\n\\tag{7}\n\\]\nTherefore\n\\[\n\\frac12+\\frac{p_{n}}{2}\\;\\le\\;P_{n}=a_{n}+p_{n}\\;\\le\\;\\frac12+p_{n}.\n\\tag{8}\n\\]\n\n\\smallskip\n\\emph{Bounding $p_{n}$.}\nBy Stirling’s estimate (ii), for $n\\ge3$,\n\\[\n\\frac{1}{\\sqrt{2\\pi n}}\\!\\Bigl(1-\\frac{1}{12n}\\Bigr)\n<p_{n}\n<\n\\frac{1}{\\sqrt{2\\pi n}}.\n\\tag{9}\n\\]\nBecause $1/\\sqrt{2\\pi}\\approx0.399<\\tfrac12$,\n\\[\np_{n}<\\frac{1}{\\sqrt{2\\pi n}}<\\frac{1}{2\\sqrt n},\n\\qquad n\\ge3.\n\\tag{10}\n\\]\n\n\\smallskip\n\\emph{Finishing the two–sided estimate.}\nInsert the left inequality of (9) into the left inequality of (8) and the\nright inequality of (10) into the right inequality of (8); for $n\\ge3$,\n\\[\n\\frac12+\\frac{1}{2\\sqrt{2\\pi n}}\\Bigl(1-\\frac{1}{12n}\\Bigr)\n<P_{n}\n<\\frac12+\\frac{1}{2\\sqrt n}.\n\\]\nSince\n\\[\n\\frac{1}{2\\sqrt{2\\pi}}\\Bigl(1-\\frac{1}{36}\\Bigr)\n>\\frac17,\n\\]\nthe lower bound is stronger than\n$\\dfrac12+\\dfrac{1}{7\\sqrt n}$ for $n\\ge3$, and part~(b) is proved.\n\n\\bigskip\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\textbf{Part (c).  Inequality for $S_{n}$}\n\nMultiplying the lower bound in (b) by $e^{n}$ yields\n\\[\nS_{n}=e^{n}P_{n}\n>\ne^{n}\\!\\left(\\frac12+\\frac{1}{7\\sqrt n}\\right),\n\\qquad n\\ge 3,\n\\]\nwhich is the desired estimate.\n\\hfill$\\square$\n\n\\bigskip\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.616694",
        "was_fixed": false,
        "difficulty_analysis": "1. Additional concepts.  \n   • The problem is now phrased in terms of the regularised incomplete-gamma function and the Poisson distribution, forcing the solver to recognise and exploit these advanced connections.  \n   • A central-limit-type estimate with an explicit speed of convergence is required; this entails invoking the Berry–Esseen theorem (or an equivalent quantitative local limit theorem).\n\n2. Extra layers.  \n   • Part (a) needs an argument based on the shape of the density t^{n}e^{−t}, going beyond the elementary rearrangement used in the original problem.  \n   • Part (b) demands explicit error bounds rather than a mere comparison with a fixed constant such as ½ or ⅓.  Handling the constant and checking its size for all n require non-trivial care.  \n\n3. Greater technical load.  \n   • Passing from inequalities with a fixed numerical constant to asymptotically sharp √n-dependent bounds involves deeper probability theory and Stirling-type approximations.  \n   • The monotonicity proof, although elementary in principle, now operates at the level of incomplete-gamma integrals rather than simple sums, increasing the analytical sophistication.\n\n4. Overall escalation.  \n   The original task was to prove an inequality with a universal constant.  \n   The enhanced variant asks for:  \n      – monotonicity,  \n      – a limiting value,  \n      – an explicit O(n^{−½}) error term with a concrete constant,  \n   thereby multiplying both the conceptual breadth and the technical depth that the solver must master."
      }
    },
    "original_kernel_variant": {
      "question": "Let  \n\\[\nS_{n}:=\\sum_{k=0}^{\\,n}\\frac{n^{k}}{k!},\\qquad  \nP_{n}:=e^{-n}S_{n}=\\mathbb P\\!\\left\\{X_{n}\\le n\\right\\},\n\\qquad X_{n}\\sim\\text{\\rm Poisson}(n),\\qquad n\\in\\mathbb N .\n\\]\n\n(a)  Prove that the sequence $\\bigl(P_{n}\\bigr)_{n\\ge 0}$ is strictly decreasing and that  \n\\[\n\\lim_{n\\to\\infty}P_{n}=\\frac12 .\n\\]\n\n(b)  Show the quantitative two–sided bound  \n\\[\n\\boxed{\\;\n\\frac12+\\frac{1}{7\\sqrt n}\n\\;<\\;\nP_{n}\n\\;<\\;\n\\frac12+\\frac{1}{2\\sqrt n}\n\\;}\\qquad(n\\ge 3).\n\\]\n\n(c)  Deduce the sharpened estimate  \n\\[\nS_{n}>e^{n}\\!\n\\left(\n\\frac12+\\frac{1}{7\\sqrt n}\n\\right)\n\\qquad(n\\ge 3).\n\\]\n\nYou may use without proof  \n\n(i)  the Gamma–integral $\\Gamma(m+1)=\\displaystyle\\int_{0}^{\\infty}t^{m}e^{-t}\\,dt$,  \n\n(ii) Stirling’s two–sided estimate  \n\\[\n\\sqrt{2\\pi}\\,m^{\\,m+\\frac12}e^{-m}\n<\nm!\n<\n\\sqrt{2\\pi}\\,m^{\\,m+\\frac12}e^{-m}\\exp\\!\\bigl(\\tfrac{1}{12m}\\bigr)\n\\qquad(m\\ge 1),\n\\]\n\n(iii) the fact that the Poisson probability mass function is unimodal, its maximum being attained at $k=n$ (and also at $k=n-1$ when $n\\ge1$).\n\n\\bigskip",
      "solution": "\\textbf{Step 0.  Probabilistic reformulation}\n\nFor $X_{n}\\sim\\text{\\rm Poisson}(n)$,\n\\[\nP_{n}=e^{-n}\\sum_{k=0}^{n}\\frac{n^{k}}{k!}\n      =\\mathbb P\\!\\bigl\\{X_{n}\\le n\\bigr\\},\n\\qquad\nS_{n}=e^{n}P_{n}.\n\\]\n\n\\bigskip\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\textbf{Step 1.  A \\emph{correct} integral representation}\n\nDenote by  \n\\[\n\\Gamma(s,x)=\\int_{x}^{\\infty}t^{s-1}e^{-t}\\,dt\\qquad(s>0,\\;x\\ge0)\n\\]\nthe \\emph{upper} incomplete Gamma function.  \nSince $\\Gamma(n+1)=n!$, changing the dummy letter gives  \n\\[\nP_{n}\n  =e^{-n}\\sum_{k=0}^{n}\\frac{n^{k}}{k!}\n  =\\frac{\\Gamma(n+1,n)}{n!}\n  =\\frac{1}{n!}\\int_{n}^{\\infty}t^{n}e^{-t}\\,dt.\n\\tag{1}\n\\]\n\n\\bigskip\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\textbf{Part (a).  Strict decrease and limit}\n\n\\emph{(i)  Strict decrease.}\nUsing (1) for $n$ and $n+1$ we obtain\n\\[\nP_{n}-P_{n+1}\n   =\\frac{1}{n!}\\!\\int_{n}^{\\infty}\\!t^{n}e^{-t}\\,dt\n    -\\frac{1}{(n+1)!}\\!\\int_{\\,n+1}^{\\infty}\\!t^{\\,n+1}e^{-t}\\,dt\n   =\\frac{1}{(n+1)!}\\Bigl[I_{n}-n^{\\,n+1}e^{-n}\\Bigr],\n\\]\nwhere  \n\\[\nI_{n}:=\\int_{n}^{\\,n+1}t^{\\,n+1}e^{-t}\\,dt.\n\\tag{2}\n\\]\n\n\\emph{Claim.}  For every $n\\ge0$ one has $I_{n}>n^{\\,n+1}e^{-n}$.\n\n\\emph{Proof of the claim.}\nPut $t=n+s$ with $0\\le s\\le1$; then\n\\[\nI_{n}=n^{\\,n+1}e^{-n}\\!\\int_{0}^{1}\n          \\Bigl(1+\\frac{s}{n}\\Bigr)^{n+1}e^{-s}\\,ds.\n\\]\nDefine  \n\\[\nf_{n}(s):=\\Bigl(1+\\frac{s}{n}\\Bigr)^{n+1}e^{-s},\\qquad 0\\le s\\le1 .\n\\]\nA simple derivative computation gives\n\\[\nf_{n}'(s)=f_{n}(s)\\,\\frac{1-s}{n+s}\\;>\\;0\n\\qquad(0<s<1),\n\\]\nso $f_{n}$ is \\emph{strictly increasing} on $[0,1]$ and $f_{n}(0)=1$.  \nHence $f_{n}(s)>1$ for $0<s\\le1$, and\n\\[\nI_{n}=n^{\\,n+1}e^{-n}\\!\\int_{0}^{1}f_{n}(s)\\,ds\n     >n^{\\,n+1}e^{-n}\\!\\int_{0}^{1}1\\,ds\n     =n^{\\,n+1}e^{-n}.\n\\]\nThe claim is proved.\n\nBecause the front factor $1/(n+1)!$ is positive, the bracket in (2) is\npositive, and\n\\[\nP_{n}>P_{n+1}\\qquad(n\\ge0).\n\\]\nStrict decrease is now established.\n\n\\smallskip\n\\emph{(ii)  The limit.}\nStandardising $X_{n}$,\n\\[\nZ_{n}:=\\frac{X_{n}-n}{\\sqrt n},\n\\]\nthe classical Central Limit Theorem yields $Z_{n}\\Longrightarrow N(0,1)$.\nSince $P_{n}=\\mathbb P\\{Z_{n}\\le0\\}$ and the Gaussian limit is symmetric,\n\\[\n\\lim_{n\\to\\infty}P_{n}=\\Phi(0)=\\frac12.\n\\]\n\n\\bigskip\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\textbf{Part (b).  A two–sided $\\sqrt n$–law}\n\nSet\n\\[\np_{n}:=\\mathbb P\\!\\bigl\\{X_{n}=n\\bigr\\}\n      =e^{-n}\\frac{n^{n}}{n!},\\qquad\na_{n}:=\\mathbb P\\!\\bigl\\{X_{n}\\le n-1\\bigr\\},\\qquad\nc_{n}:=\\mathbb P\\!\\bigl\\{X_{n}\\ge n+1\\bigr\\},\n\\]\nso that\n\\[\nP_{n}=a_{n}+p_{n},\\qquad\n1=a_{n}+p_{n}+c_{n}.\n\\]\n\n\\medskip\n\\emph{Lemma 1 (right tail dominates the far left).}\nFor every $n\\ge1$ and $h\\ge0$,\n\\[\n\\mathbb P\\!\\bigl\\{X_{n}=n+h\\bigr\\}\\;\\ge\\;\n\\mathbb P\\!\\bigl\\{X_{n}=n-1-h\\bigr\\}.\n\\tag{3}\n\\]\n\n\\emph{Proof.}\nUsing the explicit density,\n\\[\nR_{h}:=\n\\frac{\\mathbb P\\{X_{n}=n+h\\}}{\\mathbb P\\{X_{n}=n-1-h\\}}\n  =n^{\\,2h+1}\\frac{(n-1-h)!}{(n+h)!}\n  =\\prod_{j=0}^{2h}\\frac{n}{n-h+j}.\n\\]\nPair the symmetric factors:\n\\[\n(n-h+j)\\,(n+h-j)\\le n^{2}\\qquad(0\\le j\\le h),\n\\]\nso each product of two consecutive denominators does not exceed $n^{2}$\nand therefore the whole denominator does not exceed the numerator\n$n^{\\,2h+1}$.  \nHence $R_{h}\\ge1$, proving (3).\n\nSumming (3) over $h\\ge0$ gives\n\\[\np_{n}+c_{n}\\;\\ge\\;a_{n}.\n\\tag{4}\n\\]\n\n\\smallskip\n\\emph{Lemma 2 (left side dominates the symmetric right).}\nFor every $n\\ge1$ and $h\\ge1$,\n\\[\n\\mathbb P\\!\\bigl\\{X_{n}=n-h\\bigr\\}\\;\\ge\\;\n\\mathbb P\\!\\bigl\\{X_{n}=n+h\\bigr\\}.\n\\tag{5}\n\\]\n\\emph{Proof.}\nThe Poisson mass function is strictly increasing up to $k=n$ and\nstrictly decreasing afterwards (Fact (iii)), whence (5).\\hfill$\\square$\n\nSumming (5) over $h\\ge1$ yields\n\\[\na_{n}\\;\\ge\\;c_{n}.\n\\tag{6}\n\\]\n\n\\smallskip\n\\emph{Consequences of the two lemmas.}\nFrom (4) we get $a_{n}\\le p_{n}+c_{n}$, while (6) implies $c_{n}\\le a_{n}$.\nCombining,\n\\[\na_{n}\\le\\frac12,\n\\qquad\na_{n}\\ge\\frac{1-p_{n}}{2}.\n\\tag{7}\n\\]\nTherefore\n\\[\n\\frac12+\\frac{p_{n}}{2}\\;\\le\\;P_{n}=a_{n}+p_{n}\\;\\le\\;\\frac12+p_{n}.\n\\tag{8}\n\\]\n\n\\smallskip\n\\emph{Bounding $p_{n}$.}\nBy Stirling’s estimate (ii), for $n\\ge3$,\n\\[\n\\frac{1}{\\sqrt{2\\pi n}}\\!\\Bigl(1-\\frac{1}{12n}\\Bigr)\n<p_{n}\n<\n\\frac{1}{\\sqrt{2\\pi n}}.\n\\tag{9}\n\\]\nBecause $1/\\sqrt{2\\pi}\\approx0.399<\\tfrac12$,\n\\[\np_{n}<\\frac{1}{\\sqrt{2\\pi n}}<\\frac{1}{2\\sqrt n},\n\\qquad n\\ge3.\n\\tag{10}\n\\]\n\n\\smallskip\n\\emph{Finishing the two–sided estimate.}\nInsert the left inequality of (9) into the left inequality of (8) and the\nright inequality of (10) into the right inequality of (8); for $n\\ge3$,\n\\[\n\\frac12+\\frac{1}{2\\sqrt{2\\pi n}}\\Bigl(1-\\frac{1}{12n}\\Bigr)\n<P_{n}\n<\\frac12+\\frac{1}{2\\sqrt n}.\n\\]\nSince\n\\[\n\\frac{1}{2\\sqrt{2\\pi}}\\Bigl(1-\\frac{1}{36}\\Bigr)\n>\\frac17,\n\\]\nthe lower bound is stronger than\n$\\dfrac12+\\dfrac{1}{7\\sqrt n}$ for $n\\ge3$, and part~(b) is proved.\n\n\\bigskip\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\textbf{Part (c).  Inequality for $S_{n}$}\n\nMultiplying the lower bound in (b) by $e^{n}$ yields\n\\[\nS_{n}=e^{n}P_{n}\n>\ne^{n}\\!\\left(\\frac12+\\frac{1}{7\\sqrt n}\\right),\n\\qquad n\\ge 3,\n\\]\nwhich is the desired estimate.\n\\hfill$\\square$\n\n\\bigskip\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.493079",
        "was_fixed": false,
        "difficulty_analysis": "1. Additional concepts.  \n   • The problem is now phrased in terms of the regularised incomplete-gamma function and the Poisson distribution, forcing the solver to recognise and exploit these advanced connections.  \n   • A central-limit-type estimate with an explicit speed of convergence is required; this entails invoking the Berry–Esseen theorem (or an equivalent quantitative local limit theorem).\n\n2. Extra layers.  \n   • Part (a) needs an argument based on the shape of the density t^{n}e^{−t}, going beyond the elementary rearrangement used in the original problem.  \n   • Part (b) demands explicit error bounds rather than a mere comparison with a fixed constant such as ½ or ⅓.  Handling the constant and checking its size for all n require non-trivial care.  \n\n3. Greater technical load.  \n   • Passing from inequalities with a fixed numerical constant to asymptotically sharp √n-dependent bounds involves deeper probability theory and Stirling-type approximations.  \n   • The monotonicity proof, although elementary in principle, now operates at the level of incomplete-gamma integrals rather than simple sums, increasing the analytical sophistication.\n\n4. Overall escalation.  \n   The original task was to prove an inequality with a universal constant.  \n   The enhanced variant asks for:  \n      – monotonicity,  \n      – a limiting value,  \n      – an explicit O(n^{−½}) error term with a concrete constant,  \n   thereby multiplying both the conceptual breadth and the technical depth that the solver must master."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}