{
  "index": "2015-B-6",
  "type": "NT",
  "tag": [
    "NT",
    "COMB",
    "ANA"
  ],
  "difficulty": "",
  "question": "For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1, \\sqrt{2k})$. Evaluate\n\\[\n\\sum_{k=1}^\\infty (-1)^{k-1} \\frac{A(k)}{k}.\n\\]\n\n\\end{itemize}\n\n\\end{document}",
  "solution": "(from \\url{artofproblemsolving.com})\nWe will prove that the sum converges to $\\pi^2/16$.\nNote first that the sum does not converge absolutely, so we are not free to rearrange it arbitrarily. For that matter, the standard alternating sum test does not apply because the absolute values of the terms does not decrease to 0, so even the convergence of the sum must be established by hand.\n\nSetting these issues aside momentarily, note that\nthe elements of the set counted by $A(k)$ are those odd positive integers $d$ for which $m = k/d$ is also an integer and $d < \\sqrt{2dm}$; if we write $d = 2\\ee-1$, then the condition on $m$ reduces to $m \\geq \\ee$. In other words, the original sum equals\n\\[\nS_1 := \\sum_{k=1}^\\infty \\sum_{{\\ee \\geq 1, m \\geq \\ee}\\atop{k = m(2\\ee-1)}} \\frac{(-1)^{m-1}}{m(2\\ee-1)},\n\\]\nand we would like to rearrange this to\n\\[\nS_2 := \\sum_{\\ee=1}^\\infty \\frac{1}{2\\ee-1} \\sum_{m=\\ee}^\\infty \\frac{(-1)^{m-1}}{m},\n\\]\nin which both sums converge by the alternating sum test. In fact a bit more is true:\nwe have\n\\[\n\\left| \\sum_{m=\\ee}^\\infty \\frac{(-1)^{m-1}}{m} \\right| < \\frac{1}{\\ee},\n\\]\nso the outer sum converges absolutely.\nIn particular, $S_2$ is the limit of the truncated sums\n\\[\nS_{2,n} = \\sum_{\\ee(2\\ee-1) \\leq n} \\frac{1}{2\\ee-1} \\sum_{m=\\ee}^\\infty \\frac{(-1)^{m-1}}{m}.\n\\]\nTo see that $S_1$ converges to the same value as $S_2$, write\n\\[\nS_{2,n} - \\sum_{k=1}^n (-1)^{k-1} \\frac{A(k)}{k} =\n\\sum_{\\ee(2\\ee-1) \\leq n} \\frac{1}{2\\ee-1} \\sum_{m=\\lfloor \\frac{n}{2\\ee-1}+1 \\rfloor}^\\infty\n\\frac{(-1)^{m-1}}{m}.\n\\]\nThe expression on the right is bounded above in absolute value by the sum $\\sum_{\\ee(2\\ee-1) \\leq n} \\frac{1}{n}$, in which the number of summands is\n%at most $\\sqrt{n/2}$ and so the total is bounded by $1/\\sqrt{2n}$.\nat most $\\sqrt{n}$ (since $\\sqrt{n}(2\\sqrt{n}-1)\\geq n$), and so the total is bounded above by $1/\\sqrt{n}$.\nHence the difference converges to zero as $n \\to \\infty$; that is, $S_1$ converges and equals $S_2$.\n\nWe may thus focus hereafter on computing $S_2$. We begin by writing\n\\[\nS_2 = \\sum_{\\ee=1}^\\infty \\frac{1}{2\\ee-1} \\sum_{m=\\ee}^\\infty (-1)^{m-1} \\int_0^1 t^{m-1}\\,dt.\n\\]\nOur next step will be to interchange the inner sum and the integral, but again this requires some justification.\n\\begin{lemma}\nLet $f_0, f_1, \\dots$ be a sequence of continuous functions on $[0,1]$ such that for each $x \\in [0,1]$, we have\n\\[\nf_0(x) \\geq f_1(x) \\geq \\cdots \\geq 0.\n\\]\nThen\n\\[\n\\sum_{n=0}^\\infty (-1)^n \\int_0^1 f_n(t)\\,dt = \\int_0^1 \\left( \\sum_{n=0}^\\infty (-1)^n f_n(t) \\right)\\,dt\n\\]\nprovided that both sums converge.\n\\end{lemma}\n\\begin{proof}\nPut $g_n(t) = f_{2n}(t) - f_{2n+1}(t) \\geq 0$; we may then rewrite the desired equality as\n\\[\n\\sum_{n=0}^\\infty \\int_0^1 g_n(t) \\,dt = \\int_0^1 \\left( \\sum_{n=0}^\\infty g_n(t) \\right)\\,dt,\n\\]\nwhich is a case of the Lebesgue monotone convergence theorem.\n\\end{proof}\nBy Lemma~1, we have\n\\begin{align*}\nS_2 &= \\sum_{\\ee=1}^\\infty \\frac{1}{2\\ee-1} \\int_0^1 \\left( \\sum_{m=\\ee}^\\infty (-1)^{m-1} t^{m-1} \\right) \\,dt \\\\\n&=  \\sum_{\\ee=1}^\\infty \\frac{1}{2\\ee-1} \\int_0^1 \\frac{(-t)^{\\ee-1}}{1+t}  \\,dt.\n\\end{align*}\nSince the outer sum is absolutely convergent, we may freely interchange it with the integral:\n\\begin{align*}\nS_2 &=  \\int_0^1 \\left(\n\\sum_{\\ee=1}^\\infty \\frac{1}{2\\ee-1}  \\frac{(-t)^{\\ee-1}}{1+t} \\right)\\,dt \\\\\n&= \\int_0^1 \\frac{1}{\\sqrt{t}(1+t)} \\left( \\sum_{\\ee=1}^\\infty \\frac{(-1)^{\\ee-1} t^{\\ee-1/2}}{2\\ee-1} \\right) \\,dt \\\\\n&= \\int_0^1 \\frac{1}{\\sqrt{t}(1+t)} \\arctan(\\sqrt{t})\\,dt \\\\\n&= \\int_0^1 \\frac{2}{1+u^2} \\arctan(u)\\,du \\qquad (u = \\sqrt{t}) \\\\\n&= \\arctan(1)^2 - \\arctan(0)^2 = \\frac{\\pi^2}{16}.\n\\end{align*}\n\n\\end{itemize}\n\\end{document}",
  "vars": [
    "k",
    "d",
    "m",
    "n",
    "t",
    "u",
    "x",
    "g_n",
    "f_n"
  ],
  "params": [
    "A",
    "S_1",
    "S_2",
    "S_2,n"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "k": "indexvar",
        "d": "divisvar",
        "m": "multivar",
        "n": "truncvar",
        "t": "paramvar",
        "u": "sqrtvar",
        "x": "realvar",
        "g_n": "seriesg",
        "f_n": "seriesf",
        "A": "oddiviscount",
        "S_1": "firstseries",
        "S_2": "secondseries",
        "S_2,n": "secondtruncate"
      },
      "question": "For each positive integer $indexvar$, let $oddiviscount(indexvar)$ be the number of odd divisors of $indexvar$ in the interval $[1, \\sqrt{2\\,indexvar})$. Evaluate\n\\[\n\\sum_{indexvar=1}^{\\infty} (-1)^{indexvar-1} \\frac{oddiviscount(indexvar)}{indexvar}.\n\\]",
      "solution": "We will prove that the sum converges to $\\pi^2/16$.\nNote first that the sum does not converge absolutely, so we are not free to rearrange it arbitrarily. For that matter, the standard alternating sum test does not apply because the absolute values of the terms do not decrease to $0$, so even the convergence of the sum must be established by hand.\n\nSetting these issues aside momentarily, note that the elements of the set counted by $oddiviscount(indexvar)$ are those odd positive integers $divisvar$ for which $multivar = indexvar/divisvar$ is also an integer and $divisvar < \\sqrt{2\\,divisvar\\,multivar}$; if we write $divisvar = 2\\ee-1$, then the condition on $multivar$ reduces to $multivar \\geq \\ee$. In other words, the original sum equals\n\\[\nfirstseries := \\sum_{indexvar=1}^{\\infty} \\sum_{\\substack{\\ee \\ge 1,\\\\ multivar \\ge \\ee} \\atop {indexvar = multivar(2\\ee-1)}} \\frac{(-1)^{multivar-1}}{multivar(2\\ee-1)},\n\\]\nand we would like to rearrange this to\n\\[\nsecondseries := \\sum_{\\ee=1}^{\\infty} \\frac{1}{2\\ee-1} \\sum_{multivar=\\ee}^{\\infty} \\frac{(-1)^{multivar-1}}{multivar},\n\\]\nin which both sums converge by the alternating-sum test. In fact a bit more is true:\n\\[\n\\Bigl|\\,\\sum_{multivar=\\ee}^{\\infty} \\frac{(-1)^{multivar-1}}{multivar}\\,\\Bigr| < \\frac{1}{\\ee},\n\\]\nso the outer sum converges absolutely. In particular, $secondseries$ is the limit of the truncated sums\n\\[\nsecondtruncate = \\sum_{\\ee(2\\ee-1) \\le \\truncvar} \\frac{1}{2\\ee-1} \\sum_{multivar=\\ee}^{\\infty} \\frac{(-1)^{multivar-1}}{multivar}.\n\\]\nTo see that $firstseries$ converges to the same value as $secondseries$, write\n\\[\nsecondtruncate \n- \\sum_{indexvar=1}^{\\truncvar} (-1)^{indexvar-1} \\frac{oddiviscount(indexvar)}{indexvar}\n= \\sum_{\\ee(2\\ee-1) \\le \\truncvar} \\frac{1}{2\\ee-1}\n \\sum_{multivar=\\lfloor \\frac{\\truncvar}{2\\ee-1}+1 \\rfloor}^{\\infty} \\frac{(-1)^{multivar-1}}{multivar}.\n\\]\nThe expression on the right is bounded above in absolute value by the sum $\\sum_{\\ee(2\\ee-1) \\le \\truncvar} \\frac{1}{\\truncvar}$, in which the number of summands is at most $\\sqrt{\\truncvar}$ (since $\\sqrt{\\truncvar}\\,(2\\sqrt{\\truncvar}-1) \\ge \\truncvar$), and so the total is bounded above by $1/\\sqrt{\\truncvar}$. Hence the difference converges to $0$ as $\\truncvar \\to \\infty$; that is, $firstseries$ converges and equals $secondseries$.\n\nWe may thus focus hereafter on computing $secondseries$. We begin by writing\n\\[\nsecondseries \n= \\sum_{\\ee=1}^{\\infty} \\frac{1}{2\\ee-1} \\sum_{multivar=\\ee}^{\\infty} (-1)^{multivar-1} \\int_{0}^{1} paramvar^{multivar-1}\\,dparamvar.\n\\]\nOur next step will be to interchange the inner sum and the integral, but again this requires some justification.\n\n\\begin{lemma}\nLet $seriesf_0, seriesf_1,\\dots$ be a sequence of continuous functions on $[0,1]$ such that for each $realvar \\in [0,1]$,\n\\[\nseriesf_0(realvar) \\ge seriesf_1(realvar) \\ge \\cdots \\ge 0.\n\\]\nThen\n\\[\n\\sum_{\\truncvar=0}^{\\infty} (-1)^{\\truncvar} \\int_{0}^{1} seriesf_{\\truncvar}(paramvar)\\,dparamvar \n= \\int_{0}^{1} \\Bigl(\\sum_{\\truncvar=0}^{\\infty} (-1)^{\\truncvar} seriesf_{\\truncvar}(paramvar)\\Bigr)\\,dparamvar,\n\\]\nprovided that both sums converge.\n\\end{lemma}\n\\begin{proof}\nPut $seriesg_{\\truncvar}(paramvar)=seriesf_{2\\truncvar}(paramvar)-seriesf_{2\\truncvar+1}(paramvar)\\ge0$; we may then rewrite the desired equality as\n\\[\n\\sum_{\\truncvar=0}^{\\infty} \\int_{0}^{1} seriesg_{\\truncvar}(paramvar)\\,dparamvar \n= \\int_{0}^{1} \\Bigl(\\sum_{\\truncvar=0}^{\\infty} seriesg_{\\truncvar}(paramvar)\\Bigr)\\,dparamvar,\n\\]\nwhich is a case of the Lebesgue monotone-convergence theorem.\n\\end{proof}\n\nBy Lemma 1,\n\\begin{align*}\nsecondseries &= \\sum_{\\ee=1}^{\\infty} \\frac{1}{2\\ee-1} \\int_{0}^{1} \\Bigl(\\sum_{multivar=\\ee}^{\\infty} (-1)^{multivar-1} paramvar^{multivar-1}\\Bigr)\\,dparamvar\\\\\n&= \\sum_{\\ee=1}^{\\infty} \\frac{1}{2\\ee-1} \\int_{0}^{1} \\frac{(-paramvar)^{\\ee-1}}{1+paramvar}\\,dparamvar.\n\\end{align*}\nSince the outer sum is absolutely convergent, we may freely interchange it with the integral:\n\\begin{align*}\nsecondseries &= \\int_{0}^{1} \\Bigl(\\sum_{\\ee=1}^{\\infty} \\frac{1}{2\\ee-1} \\frac{(-paramvar)^{\\ee-1}}{1+paramvar}\\Bigr)\\,dparamvar\\\\\n&= \\int_{0}^{1} \\frac{1}{\\sqrt{paramvar}\\,(1+paramvar)} \\Bigl(\\sum_{\\ee=1}^{\\infty} \\frac{(-1)^{\\ee-1} paramvar^{\\ee-1/2}}{2\\ee-1}\\Bigr)\\,dparamvar\\\\\n&= \\int_{0}^{1} \\frac{1}{\\sqrt{paramvar}\\,(1+paramvar)}\\arctan(\\sqrt{paramvar})\\,dparamvar\\\\\n&= \\int_{0}^{1} \\frac{2}{1+sqrtvar^{2}}\\arctan(sqrtvar)\\,d sqrtvar \\qquad (sqrtvar=\\sqrt{paramvar})\\\\\n&= \\arctan(1)^{2}-\\arctan(0)^{2}=\\frac{\\pi^{2}}{16}.\n\\end{align*}"
    },
    "descriptive_long_confusing": {
      "map": {
        "k": "kangaroo",
        "d": "daffodil",
        "m": "marigold",
        "n": "nectarine",
        "t": "tablespoon",
        "u": "underpass",
        "x": "xylophone",
        "g_n": "grenadine",
        "f_n": "feathered",
        "A": "appetite",
        "S_1": "sandstone",
        "S_2": "sunflower",
        "S_2,n": "sunflowerpetal"
      },
      "question": "For each positive integer $kangaroo$, let $appetite(kangaroo)$ be the number of odd divisors of $kangaroo$ in the interval $[1, \\sqrt{2kangaroo})$. Evaluate\n\\[\n\\sum_{kangaroo=1}^{\\infty} (-1)^{kangaroo-1} \\frac{appetite(kangaroo)}{kangaroo}.\n\\]",
      "solution": "We will prove that the sum converges to $\\pi^2/16$.\nNote first that the sum does not converge absolutely, so we are not free to rearrange it arbitrarily. For that matter, the standard alternating sum test does not apply because the absolute values of the terms does not decrease to 0, so even the convergence of the sum must be established by hand.\n\nSetting these issues aside momentarily, note that\nthe elements of the set counted by $appetite(kangaroo)$ are those odd positive integers $daffodil$ for which $marigold = kangaroo/daffodil$ is also an integer and $daffodil < \\sqrt{2daffodil marigold}$; if we write $daffodil = 2\\ee-1$, then the condition on $marigold$ reduces to $marigold \\geq \\ee$. In other words, the original sum equals\n\\[\nsandstone := \\sum_{kangaroo=1}^{\\infty} \\sum_{{\\ee \\geq 1, marigold \\geq \\ee}\\atop{kangaroo = marigold(2\\ee-1)}} \\frac{(-1)^{marigold-1}}{marigold(2\\ee-1)},\n\\]\nand we would like to rearrange this to\n\\[\nsunflower := \\sum_{\\ee=1}^{\\infty} \\frac{1}{2\\ee-1} \\sum_{marigold=\\ee}^{\\infty} \\frac{(-1)^{marigold-1}}{marigold},\n\\]\nin which both sums converge by the alternating sum test. In fact a bit more is true:\nwe have\n\\[\n\\left| \\sum_{marigold=\\ee}^{\\infty} \\frac{(-1)^{marigold-1}}{marigold} \\right| < \\frac{1}{\\ee},\n\\]\nso the outer sum converges absolutely.\nIn particular, sunflower is the limit of the truncated sums\n\\[\nsunflowerpetal = \\sum_{\\ee(2\\ee-1) \\leq nectarine} \\frac{1}{2\\ee-1} \\sum_{marigold=\\ee}^{\\infty} \\frac{(-1)^{marigold-1}}{marigold}.\n\\]\nTo see that sandstone converges to the same value as sunflower, write\n\\[\nsunflowerpetal - \\sum_{kangaroo=1}^{nectarine} (-1)^{kangaroo-1} \\frac{appetite(kangaroo)}{kangaroo} =\n\\sum_{\\ee(2\\ee-1) \\leq nectarine} \\frac{1}{2\\ee-1} \\sum_{marigold=\\lfloor \\frac{nectarine}{2\\ee-1}+1 \\rfloor}^{\\infty}\n\\frac{(-1)^{marigold-1}}{marigold}.\n\\]\nThe expression on the right is bounded above in absolute value by the sum $\\sum_{\\ee(2\\ee-1) \\leq nectarine} \\frac{1}{nectarine}$, in which the number of summands is\n%at most $\\sqrt{nectarine/2}$ and so the total is bounded by $1/\\sqrt{2nectarine}$.\nat most $\\sqrt{nectarine}$ (since $\\sqrt{nectarine}(2\\sqrt{nectarine}-1)\\geq nectarine$), and so the total is bounded above by $1/\\sqrt{nectarine}$.\nHence the difference converges to zero as $nectarine \\to \\infty$; that is, sandstone converges and equals sunflower.\n\nWe may thus focus hereafter on computing sunflower. We begin by writing\n\\[\nsunflower = \\sum_{\\ee=1}^{\\infty} \\frac{1}{2\\ee-1} \\sum_{marigold=\\ee}^{\\infty} (-1)^{marigold-1} \\int_0^1 tablespoon^{marigold-1}\\,d tablespoon.\n\\]\nOur next step will be to interchange the inner sum and the integral, but again this requires some justification.\n\\begin{lemma}\nLet $f_0, f_1, \\dots$ be a sequence of continuous functions on $[0,1]$ such that for each $xylophone \\in [0,1]$, we have\n\\[\nf_0(xylophone) \\geq f_1(xylophone) \\geq \\cdots \\geq 0.\n\\]\nThen\n\\[\n\\sum_{nectarine=0}^{\\infty} (-1)^{nectarine} \\int_0^1 feathered(tablespoon)\\,d tablespoon = \\int_0^1 \\left( \\sum_{nectarine=0}^{\\infty} (-1)^{nectarine} feathered(tablespoon) \\right)\\,d tablespoon\n\\]\nprovided that both sums converge.\n\\end{lemma}\n\\begin{proof}\nPut grenadine_{nectarine}(tablespoon) = f_{2nectarine}(tablespoon) - f_{2nectarine+1}(tablespoon) \\geq 0; we may then rewrite the desired equality as\n\\[\n\\sum_{nectarine=0}^{\\infty} \\int_0^1 grenadine_{nectarine}(tablespoon) \\,d tablespoon = \\int_0^1 \\left( \\sum_{nectarine=0}^{\\infty} grenadine_{nectarine}(tablespoon) \\right)\\,d tablespoon,\n\\]\nwhich is a case of the Lebesgue monotone convergence theorem.\n\\end{proof}\nBy Lemma~1, we have\n\\begin{align*}\nsunflower &= \\sum_{\\ee=1}^{\\infty} \\frac{1}{2\\ee-1} \\int_0^1 \\left( \\sum_{marigold=\\ee}^{\\infty} (-1)^{marigold-1} tablespoon^{marigold-1} \\right) \\,d tablespoon \\\\\n&=  \\sum_{\\ee=1}^{\\infty} \\frac{1}{2\\ee-1} \\int_0^1 \\frac{(-tablespoon)^{\\ee-1}}{1+tablespoon}  \\,d tablespoon.\n\\end{align*}\nSince the outer sum is absolutely convergent, we may freely interchange it with the integral:\n\\begin{align*}\nsunflower &=  \\int_0^1 \\left(\n\\sum_{\\ee=1}^{\\infty} \\frac{1}{2\\ee-1}  \\frac{(-tablespoon)^{\\ee-1}}{1+tablespoon} \\right)\\,d tablespoon \\\\\n&= \\int_0^1 \\frac{1}{\\sqrt{tablespoon}(1+tablespoon)} \\left( \\sum_{\\ee=1}^{\\infty} \\frac{(-1)^{\\ee-1} tablespoon^{\\ee-1/2}}{2\\ee-1} \\right) \\,d tablespoon \\\\\n&= \\int_0^1 \\frac{1}{\\sqrt{tablespoon}(1+tablespoon)} \\arctan(\\sqrt{tablespoon})\\,d tablespoon \\\\\n&= \\int_0^1 \\frac{2}{1+underpass^2} \\arctan(underpass)\\,d underpass \\qquad (underpass = \\sqrt{tablespoon}) \\\\\n&= \\arctan(1)^2 - \\arctan(0)^2 = \\frac{\\pi^2}{16}.\n\\end{align*}"
    },
    "descriptive_long_misleading": {
      "map": {
        "k": "constantvalue",
        "d": "multipleeven",
        "m": "productvalue",
        "n": "infinitevalue",
        "t": "constantunity",
        "u": "staticvalue",
        "x": "immobilept",
        "g_n": "negativeseq",
        "f_n": "constantseq",
        "A": "evenmultiplier",
        "S_1": "differencestart",
        "S_2": "differencefinal",
        "S_2,n": "differencepartial"
      },
      "question": "For each positive integer $constantvalue$, let $evenmultiplier(constantvalue)$ be the number of odd divisors of $constantvalue$ in the interval $[1, \\sqrt{2constantvalue})$. Evaluate\n\\[\n\\sum_{constantvalue=1}^\\infty (-1)^{constantvalue-1} \\frac{evenmultiplier(constantvalue)}{constantvalue}.\n\\]\n",
      "solution": "We will prove that the sum converges to $\\pi^2/16$.\nNote first that the sum does not converge absolutely, so we are not free to rearrange it arbitrarily. For that matter, the standard alternating sum test does not apply because the absolute values of the terms does not decrease to 0, so even the convergence of the sum must be established by hand.\n\nSetting these issues aside momentarily, note that\nthe elements of the set counted by $evenmultiplier(constantvalue)$ are those odd positive integers $multipleeven$ for which $productvalue = constantvalue/multipleeven$ is also an integer and $multipleeven < \\sqrt{2 multipleeven productvalue}$; if we write $multipleeven = 2\\ee-1$, then the condition on $productvalue$ reduces to $productvalue \\geq \\ee$. In other words, the original sum equals\n\\[\ndifferencestart := \\sum_{constantvalue=1}^\\infty \\sum_{{\\ee \\geq 1, productvalue \\geq \\ee}\\atop{constantvalue = productvalue(2\\ee-1)}} \\frac{(-1)^{productvalue-1}}{productvalue(2\\ee-1)},\n\\]\nand we would like to rearrange this to\n\\[\ndifferencefinal := \\sum_{\\ee=1}^\\infty \\frac{1}{2\\ee-1} \\sum_{productvalue=\\ee}^\\infty \\frac{(-1)^{productvalue-1}}{productvalue},\n\\]\nin which both sums converge by the alternating sum test. In fact a bit more is true:\nwe have\n\\[\n\\left| \\sum_{productvalue=\\ee}^\\infty \\frac{(-1)^{productvalue-1}}{productvalue} \\right| < \\frac{1}{\\ee},\n\\]\nso the outer sum converges absolutely.\nIn particular, $differencefinal$ is the limit of the truncated sums\n\\[\ndifferencepartial = \\sum_{\\ee(2\\ee-1) \\leq infinitevalue} \\frac{1}{2\\ee-1} \\sum_{productvalue=\\ee}^\\infty \\frac{(-1)^{productvalue-1}}{productvalue}.\n\\]\nTo see that $differencestart$ converges to the same value as $differencefinal$, write\n\\[\ndifferencepartial - \\sum_{constantvalue=1}^{infinitevalue} (-1)^{constantvalue-1} \\frac{evenmultiplier(constantvalue)}{constantvalue} =\n\\sum_{\\ee(2\\ee-1) \\leq infinitevalue} \\frac{1}{2\\ee-1} \\sum_{productvalue=\\lfloor \\frac{infinitevalue}{2\\ee-1}+1 \\rfloor}^\\infty\n\\frac{(-1)^{productvalue-1}}{productvalue}.\n\\]\nThe expression on the right is bounded above in absolute value by the sum $\\sum_{\\ee(2\\ee-1) \\leq infinitevalue} \\frac{1}{infinitevalue}$, in which the number of summands is\nat most $\\sqrt{infinitevalue}$ (since $\\sqrt{infinitevalue}(2\\sqrt{infinitevalue}-1)\\geq infinitevalue$), and so the total is bounded above by $1/\\sqrt{infinitevalue}$.\nHence the difference converges to zero as $infinitevalue \\to \\infty$; that is, $differencestart$ converges and equals $differencefinal$.\n\nWe may thus focus hereafter on computing $differencefinal$. We begin by writing\n\\[\ndifferencefinal = \\sum_{\\ee=1}^\\infty \\frac{1}{2\\ee-1} \\sum_{productvalue=\\ee}^\\infty (-1)^{productvalue-1} \\int_0^1 constantunity^{productvalue-1}\\,d constantunity.\n\\]\nOur next step will be to interchange the inner sum and the integral, but again this requires some justification.\n\\begin{lemma}\nLet $constantseq_0, constantseq_1, \\dots$ be a sequence of continuous functions on $[0,1]$ such that for each $immobilept \\in [0,1]$, we have\n\\[\nconstantseq_0(immobilept) \\geq constantseq_1(immobilept) \\geq \\cdots \\geq 0.\n\\]\nThen\n\\[\n\\sum_{\\infinitevalue=0}^\\infty (-1)^{\\infinitevalue} \\int_0^1 constantseq_{\\infinitevalue}(constantunity)\\,d constantunity = \\int_0^1 \\left( \\sum_{\\infinitevalue=0}^\\infty (-1)^{\\infinitevalue} constantseq_{\\infinitevalue}(constantunity) \\right)\\,d constantunity\n\\]\nprovided that both sums converge.\n\\end{lemma}\n\\begin{proof}\nPut $negativeseq_{\\infinitevalue}(constantunity) = constantseq_{2\\infinitevalue}(constantunity) - constantseq_{2\\infinitevalue+1}(constantunity) \\geq 0$; we may then rewrite the desired equality as\n\\[\n\\sum_{\\infinitevalue=0}^\\infty \\int_0^1 negativeseq_{\\infinitevalue}(constantunity) \\,d constantunity = \\int_0^1 \\left( \\sum_{\\infinitevalue=0}^\\infty negativeseq_{\\infinitevalue}(constantunity) \\right)\\,d constantunity,\n\\]\nwhich is a case of the Lebesgue monotone convergence theorem.\n\\end{proof}\nBy Lemma~1, we have\n\\begin{align*}\ndifferencefinal &= \\sum_{\\ee=1}^\\infty \\frac{1}{2\\ee-1} \\int_0^1 \\left( \\sum_{productvalue=\\ee}^\\infty (-1)^{productvalue-1} constantunity^{productvalue-1} \\right) \\,d constantunity \\\\\n&=  \\sum_{\\ee=1}^\\infty \\frac{1}{2\\ee-1} \\int_0^1 \\frac{(- constantunity)^{\\ee-1}}{1+constantunity}  \\,d constantunity.\n\\end{align*}\nSince the outer sum is absolutely convergent, we may freely interchange it with the integral:\n\\begin{align*}\ndifferencefinal &=  \\int_0^1 \\left(\n\\sum_{\\ee=1}^\\infty \\frac{1}{2\\ee-1}  \\frac{(- constantunity)^{\\ee-1}}{1+constantunity} \\right)\\,d constantunity \\\\\n&= \\int_0^1 \\frac{1}{\\sqrt{constantunity}(1+constantunity)} \\left( \\sum_{\\ee=1}^\\infty \\frac{(-1)^{\\ee-1} constantunity^{\\ee-1/2}}{2\\ee-1} \\right) \\,d constantunity \\\\\n&= \\int_0^1 \\frac{1}{\\sqrt{constantunity}(1+constantunity)} \\arctan(\\sqrt{constantunity})\\,d constantunity \\\\\n&= \\int_0^1 \\frac{2}{1+staticvalue^2} \\arctan(staticvalue)\\,d staticvalue \\qquad (staticvalue = \\sqrt{constantunity}) \\\\\n&= \\arctan(1)^2 - \\arctan(0)^2 = \\frac{\\pi^2}{16}.\n\\end{align*}\n"
    },
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        "n": "yplodkfj",
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        "u": "lkjhmnbv",
        "x": "poiruqwe",
        "g_n": "plorxuet",
        "f_n": "omnibqsa",
        "A": "qwertyui",
        "S_1": "asdfghjk",
        "S_2": "zxcvbnml",
        "S_2,n": "poiuytre"
      },
      "question": "For each positive integer $qzxwvtnp$, let $qwertyui(qzxwvtnp)$ be the number of odd divisors of $qzxwvtnp$ in the interval $[1, \\sqrt{2qzxwvtnp})$. Evaluate\n\\[\n\\sum_{qzxwvtnp=1}^\\infty (-1)^{qzxwvtnp-1} \\frac{qwertyui(qzxwvtnp)}{qzxwvtnp}.\n\\]",
      "solution": "(from \\url{artofproblemsolving.com})\nWe will prove that the sum converges to $\\pi^2/16$.\nNote first that the sum does not converge absolutely, so we are not free to rearrange it arbitrarily. For that matter, the standard alternating sum test does not apply because the absolute values of the terms does not decrease to 0, so even the convergence of the sum must be established by hand.\n\nSetting these issues aside momentarily, note that\nthe elements of the set counted by $qwertyui(qzxwvtnp)$ are those odd positive integers $hjgrksla$ for which $vbncxzlk = qzxwvtnp/hjgrksla$ is also an integer and $hjgrksla < \\sqrt{2hjgrksla vbncxzlk}$; if we write $hjgrksla = 2\\ee-1$, then the condition on $vbncxzlk$ reduces to $vbncxzlk \\geq \\ee$. In other words, the original sum equals\n\\[\nasdfghjk := \\sum_{qzxwvtnp=1}^\\infty \\sum_{{\\ee \\geq 1, vbncxzlk \\geq \\ee}\\atop{qzxwvtnp = vbncxzlk(2\\ee-1)}} \\frac{(-1)^{vbncxzlk-1}}{vbncxzlk(2\\ee-1)},\n\\]\nand we would like to rearrange this to\n\\[\nzxcvbnml := \\sum_{\\ee=1}^\\infty \\frac{1}{2\\ee-1} \\sum_{vbncxzlk=\\ee}^\\infty \\frac{(-1)^{vbncxzlk-1}}{vbncxzlk},\n\\]\nin which both sums converge by the alternating sum test. In fact a bit more is true:\nwe have\n\\[\n\\left| \\sum_{vbncxzlk=\\ee}^\\infty \\frac{(-1)^{vbncxzlk-1}}{vbncxzlk} \\right| < \\frac{1}{\\ee},\n\\]\nso the outer sum converges absolutely.\nIn particular, zxcvbnml is the limit of the truncated sums\n\\[\npoiuytre = \\sum_{\\ee(2\\ee-1) \\leq yplodkfj} \\frac{1}{2\\ee-1} \\sum_{vbncxzlk=\\ee}^\\infty \\frac{(-1)^{vbncxzlk-1}}{vbncxzlk}.\n\\]\nTo see that asdfghjk converges to the same value as zxcvbnml, write\n\\[\npoiuytre - \\sum_{qzxwvtnp=1}^{yplodkfj} (-1)^{qzxwvtnp-1} \\frac{qwertyui(qzxwvtnp)}{qzxwvtnp} =\n\\sum_{\\ee(2\\ee-1) \\leq yplodkfj} \\frac{1}{2\\ee-1} \\sum_{vbncxzlk=\\lfloor \\frac{yplodkfj}{2\\ee-1}+1 \\rfloor}^\\infty\n\\frac{(-1)^{vbncxzlk-1}}{vbncxzlk}.\n\\]\nThe expression on the right is bounded above in absolute value by the sum $\\sum_{\\ee(2\\ee-1) \\leq yplodkfj} \\frac{1}{yplodkfj}$, in which the number of summands is\n%at most $\\sqrt{yplodkfj/2}$ and so the total is bounded by $1/\\sqrt{2yplodkfj}$.\nat most $\\sqrt{yplodkfj}$ (since $\\sqrt{yplodkfj}(2\\sqrt{yplodkfj}-1)\\geq yplodkfj$), and so the total is bounded above by $1/\\sqrt{yplodkfj}$.\nHence the difference converges to zero as $yplodkfj \\to \\infty$; that is, asdfghjk converges and equals zxcvbnml.\n\nWe may thus focus hereafter on computing zxcvbnml. We begin by writing\n\\[\nzxcvbnml = \\sum_{\\ee=1}^\\infty \\frac{1}{2\\ee-1} \\sum_{vbncxzlk=\\ee}^\\infty (-1)^{vbncxzlk-1} \\int_0^1 abncedqr^{vbncxzlk-1}\\,d abncedqr.\n\\]\nOur next step will be to interchange the inner sum and the integral, but again this requires some justification.\n\\begin{lemma}\nLet omnibqsa_0, omnibqsa_1, \\dots be a sequence of continuous functions on $[0,1]$ such that for each poiruqwe \\in [0,1], we have\n\\[\nomnibqsa_0(poiruqwe) \\geq omnibqsa_1(poiruqwe) \\geq \\cdots \\geq 0.\n\\]\nThen\n\\[\n\\sum_{yplodkfj=0}^\\infty (-1)^{yplodkfj} \\int_0^1 omnibqsa_{yplodkfj}(abncedqr)\\,d abncedqr = \\int_0^1 \\left( \\sum_{yplodkfj=0}^\\infty (-1)^{yplodkfj} omnibqsa_{yplodkfj}(abncedqr) \\right)\\,d abncedqr\n\\]\nprovided that both sums converge.\n\\end{lemma}\n\\begin{proof}\nPut plorxuet_{yplodkfj}(abncedqr) = omnibqsa_{2yplodkfj}(abncedqr) - omnibqsa_{2yplodkfj+1}(abncedqr) \\geq 0; we may then rewrite the desired equality as\n\\[\n\\sum_{yplodkfj=0}^\\infty \\int_0^1 plorxuet_{yplodkfj}(abncedqr) \\,d abncedqr = \\int_0^1 \\left( \\sum_{yplodkfj=0}^\\infty plorxuet_{yplodkfj}(abncedqr) \\right)\\,d abncedqr,\n\\]\nwhich is a case of the Lebesgue monotone convergence theorem.\n\\end{proof}\nBy Lemma~1, we have\n\\begin{align*}\nzxcvbnml &= \\sum_{\\ee=1}^\\infty \\frac{1}{2\\ee-1} \\int_0^1 \\left( \\sum_{vbncxzlk=\\ee}^\\infty (-1)^{vbncxzlk-1} abncedqr^{vbncxzlk-1} \\right) \\,d abncedqr \\\n&=  \\sum_{\\ee=1}^\\infty \\frac{1}{2\\ee-1} \\int_0^1 \\frac{(-abncedqr)^{\\ee-1}}{1+abncedqr}  \\,d abncedqr.\n\\end{align*}\nSince the outer sum is absolutely convergent, we may freely interchange it with the integral:\n\\begin{align*}\nzxcvbnml &=  \\int_0^1 \\left(\n\\sum_{\\ee=1}^\\infty \\frac{1}{2\\ee-1}  \\frac{(-abncedqr)^{\\ee-1}}{1+abncedqr} \\right)\\,d abncedqr \\\\\n&= \\int_0^1 \\frac{1}{\\sqrt{abncedqr}(1+abncedqr)} \\left( \\sum_{\\ee=1}^\\infty \\frac{(-1)^{\\ee-1} abncedqr^{\\ee-1/2}}{2\\ee-1} \\right) \\,d abncedqr \\\\\n&= \\int_0^1 \\frac{1}{\\sqrt{abncedqr}(1+abncedqr)} \\arctan(\\sqrt{abncedqr})\\,d abncedqr \\\\\n&= \\int_0^1 \\frac{2}{1+lkjhmnbv^2} \\arctan(lkjhmnbv)\\,d lkjhmnbv \\qquad (lkjhmnbv = \\sqrt{abncedqr}) \\\\\n&= \\arctan(1)^2 - \\arctan(0)^2 = \\frac{\\pi^2}{16}.\n\\end{align*}"
    },
    "kernel_variant": {
      "question": "Let  \nχ(n)=⎧⎪⎨⎪⎩ 0             if n is even,  \n            1             if n≡1 (mod 4),  \n           –1             if n≡3 (mod 4)  \nbe the non–principal Dirichlet character modulo 4 (equivalently χ(n)=(-1)^{(n-1)/2} for odd n).  \n\nFor every positive integer k define the χ–weighted count  \n  C(k)=∑_{d∣k,\\;d odd,\\;d<√k}χ(d).  \n\nEvaluate the alternating series  \n  S = ∑_{k=1}^{∞} (-1)^{k-1} · C(k) ⁄ k.  \n\n————————————————————————————————————————",
      "solution": "1.  Rewriting the series as a double sum  \n   Because d is required to be odd, write k=d m with d odd and m≥1.  \n   The inequality d<√k is equivalent to d²<d m, i.e. m>d.  Thus\n\n      S = ∑_{d odd} χ(d)∑_{m>d} (-1)^{dm-1}/(dm)                                (1)\n\n   Since d is odd, (-1)^{d m}=(-1)^{m}; hence (-1)^{dm-1}=(-1)^{m-1}.  Equation (1) becomes\n\n      S = ∑_{d odd} χ(d)/d · T(d),   where   T(d)=∑_{m=d+1}^{∞} (-1)^{m-1}/m.   (2)\n\n   Convergence of (2):  |T(d)|≤1/(d+1), so the outer sum in (2) converges absolutely.\n\n2.  Integral representation of the alternating tail  \n   For every integer N≥0,\n\n      ∑_{m=N+1}^{∞} (-1)^{m-1}/m = ∫_{0}^{1} (-t)^{N}/(1+t) dt.                (3)\n\n   Setting N=d in (3) gives T(d)=∫_{0}^{1} (-t)^{d}/(1+t) dt.  Substitute this into (2) and interchange\n   sum and integral (justified by absolute convergence):\n\n      S = ∫_{0}^{1} \\frac{1}{1+t}\\Bigl(\\sum_{d odd} χ(d)\\,(-t)^{d}/d\\Bigr)dt.     (4)\n\n3.  The χ–Dirichlet series that appears  \n   Write d=2n+1 (n≥0).  For odd d, χ(d)=(-1)^{n}.  Hence\n\n      ∑_{d odd} χ(d)(-t)^{d}/d\n      = ∑_{n=0}^{∞} (-1)^{n} (-t)^{2n+1}/(2n+1)\n      = −∑_{n=0}^{∞} (-1)^{n} t^{2n+1}/(2n+1)                                   (5)\n\n   But the last series is –arctan t.  Therefore the bracket in (4) equals –arctan t, and\n\n      S = −∫_{0}^{1} \\frac{\\arctan t}{1+t}\\,dt.                                  (6)\n\n4.  Evaluating the integral  \n   Let I=∫_{0}^{1} arctan t/(1+t) dt.\n\n   Integration by parts:  set u=arctan t, dv=dt/(1+t).  Then du=dt/(1+t^{2}),\n   v=ln(1+t).  Hence\n\n      I = [arctan t·ln(1+t)]_{0}^{1} − ∫_{0}^{1} ln(1+t)/(1+t^{2}) dt\n        = (π/4)ln 2 − J,                                                         (7)\n\n   where J=∫_{0}^{1} ln(1+t)/(1+t^{2}) dt.\n\n   Computation of J.  Substitute t=tan θ (θ∈[0,π/4]):  \n   dt/(1+t^{2})=dθ, and 1+t=1+tan θ.  Thus\n\n      J = ∫_{0}^{π/4} ln(1+tan θ)\\,dθ.                                          (8)\n\n   Write 1+tan θ = (sin θ+cos θ)/cos θ, so\n\n      ln(1+tan θ)=ln(sin θ+cos θ)−ln cos θ.\n\n   With the identities sin θ+cos θ = √2 sin(θ+π/4) and the well–known integrals\n      ∫_{0}^{π/4} ln sin θ dθ = −(π/4)ln 2 − G/2,   \n      ∫_{0}^{π/4} ln cos θ dθ = −(π/4)ln 2 + G/2\n\n   (where G denotes Catalan’s constant), a straightforward calculation gives\n\n      J = (π/8)ln 2.                                                             (9)\n\n   Insert (9) into (7):  I = (π/4)ln 2 − (π/8)ln 2 = (π/8)ln 2.\n\n   Finally, by (6),\n\n      S = −I = −(π/8)\\,ln 2.                                                     (10)\n\n5.  Answer  \n   ∑_{k=1}^{∞} (-1)^{k-1} C(k)/k =  – π ln 2 ⁄ 8.\n\n————————————————————————————————————————",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.840417",
        "was_fixed": false,
        "difficulty_analysis": "• Weighted counting:  The problem replaces a simple divisor count by a χ–weighted count,\n   forcing the solver to recognise and manipulate a non-trivial Dirichlet character.  \n\n• More stringent bound:  The window 1≤d<√k (instead of √{2k}) changes the\n   arithmetic condition from m≥⌈d/2⌉ to m>d, yielding a more intricate lower limit in the inner sum.  \n\n• Advanced analytic tools:  \n   – Conversion of a two-variable arithmetic sum into a convergent Dirichlet series with a\n      character.  \n   – Use of an integral transform to evaluate an alternating harmonic tail for a variable\n      lower bound.  \n   – Identification of a generating function with –arctan t, tying the computation to special\n      values of L-functions.  \n   – Evaluation of the remaining integral requires Catalan’s constant, delicate sine–cosine\n      log-integrals, and careful use of symmetry identities—considerably deeper than the single\n      arctangent integral in the original problem.  \n\n• Convergence justifications now demand absolute/conditional convergence arguments for\n   Dirichlet series with characters, rather than simple alternating-series reasoning.  \n\nThese layers of number–theoretic, analytic, and special-function techniques make the\nenhanced variant substantially more challenging than both the original problem and the\ncurrent kernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Let  \nχ(n)=⎧⎪⎨⎪⎩ 0             if n is even,  \n            1             if n≡1 (mod 4),  \n           –1             if n≡3 (mod 4)  \nbe the non–principal Dirichlet character modulo 4 (equivalently χ(n)=(-1)^{(n-1)/2} for odd n).  \n\nFor every positive integer k define the χ–weighted count  \n  C(k)=∑_{d∣k,\\;d odd,\\;d<√k}χ(d).  \n\nEvaluate the alternating series  \n  S = ∑_{k=1}^{∞} (-1)^{k-1} · C(k) ⁄ k.  \n\n————————————————————————————————————————",
      "solution": "1.  Rewriting the series as a double sum  \n   Because d is required to be odd, write k=d m with d odd and m≥1.  \n   The inequality d<√k is equivalent to d²<d m, i.e. m>d.  Thus\n\n      S = ∑_{d odd} χ(d)∑_{m>d} (-1)^{dm-1}/(dm)                                (1)\n\n   Since d is odd, (-1)^{d m}=(-1)^{m}; hence (-1)^{dm-1}=(-1)^{m-1}.  Equation (1) becomes\n\n      S = ∑_{d odd} χ(d)/d · T(d),   where   T(d)=∑_{m=d+1}^{∞} (-1)^{m-1}/m.   (2)\n\n   Convergence of (2):  |T(d)|≤1/(d+1), so the outer sum in (2) converges absolutely.\n\n2.  Integral representation of the alternating tail  \n   For every integer N≥0,\n\n      ∑_{m=N+1}^{∞} (-1)^{m-1}/m = ∫_{0}^{1} (-t)^{N}/(1+t) dt.                (3)\n\n   Setting N=d in (3) gives T(d)=∫_{0}^{1} (-t)^{d}/(1+t) dt.  Substitute this into (2) and interchange\n   sum and integral (justified by absolute convergence):\n\n      S = ∫_{0}^{1} \\frac{1}{1+t}\\Bigl(\\sum_{d odd} χ(d)\\,(-t)^{d}/d\\Bigr)dt.     (4)\n\n3.  The χ–Dirichlet series that appears  \n   Write d=2n+1 (n≥0).  For odd d, χ(d)=(-1)^{n}.  Hence\n\n      ∑_{d odd} χ(d)(-t)^{d}/d\n      = ∑_{n=0}^{∞} (-1)^{n} (-t)^{2n+1}/(2n+1)\n      = −∑_{n=0}^{∞} (-1)^{n} t^{2n+1}/(2n+1)                                   (5)\n\n   But the last series is –arctan t.  Therefore the bracket in (4) equals –arctan t, and\n\n      S = −∫_{0}^{1} \\frac{\\arctan t}{1+t}\\,dt.                                  (6)\n\n4.  Evaluating the integral  \n   Let I=∫_{0}^{1} arctan t/(1+t) dt.\n\n   Integration by parts:  set u=arctan t, dv=dt/(1+t).  Then du=dt/(1+t^{2}),\n   v=ln(1+t).  Hence\n\n      I = [arctan t·ln(1+t)]_{0}^{1} − ∫_{0}^{1} ln(1+t)/(1+t^{2}) dt\n        = (π/4)ln 2 − J,                                                         (7)\n\n   where J=∫_{0}^{1} ln(1+t)/(1+t^{2}) dt.\n\n   Computation of J.  Substitute t=tan θ (θ∈[0,π/4]):  \n   dt/(1+t^{2})=dθ, and 1+t=1+tan θ.  Thus\n\n      J = ∫_{0}^{π/4} ln(1+tan θ)\\,dθ.                                          (8)\n\n   Write 1+tan θ = (sin θ+cos θ)/cos θ, so\n\n      ln(1+tan θ)=ln(sin θ+cos θ)−ln cos θ.\n\n   With the identities sin θ+cos θ = √2 sin(θ+π/4) and the well–known integrals\n      ∫_{0}^{π/4} ln sin θ dθ = −(π/4)ln 2 − G/2,   \n      ∫_{0}^{π/4} ln cos θ dθ = −(π/4)ln 2 + G/2\n\n   (where G denotes Catalan’s constant), a straightforward calculation gives\n\n      J = (π/8)ln 2.                                                             (9)\n\n   Insert (9) into (7):  I = (π/4)ln 2 − (π/8)ln 2 = (π/8)ln 2.\n\n   Finally, by (6),\n\n      S = −I = −(π/8)\\,ln 2.                                                     (10)\n\n5.  Answer  \n   ∑_{k=1}^{∞} (-1)^{k-1} C(k)/k =  – π ln 2 ⁄ 8.\n\n————————————————————————————————————————",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.643146",
        "was_fixed": false,
        "difficulty_analysis": "• Weighted counting:  The problem replaces a simple divisor count by a χ–weighted count,\n   forcing the solver to recognise and manipulate a non-trivial Dirichlet character.  \n\n• More stringent bound:  The window 1≤d<√k (instead of √{2k}) changes the\n   arithmetic condition from m≥⌈d/2⌉ to m>d, yielding a more intricate lower limit in the inner sum.  \n\n• Advanced analytic tools:  \n   – Conversion of a two-variable arithmetic sum into a convergent Dirichlet series with a\n      character.  \n   – Use of an integral transform to evaluate an alternating harmonic tail for a variable\n      lower bound.  \n   – Identification of a generating function with –arctan t, tying the computation to special\n      values of L-functions.  \n   – Evaluation of the remaining integral requires Catalan’s constant, delicate sine–cosine\n      log-integrals, and careful use of symmetry identities—considerably deeper than the single\n      arctangent integral in the original problem.  \n\n• Convergence justifications now demand absolute/conditional convergence arguments for\n   Dirichlet series with characters, rather than simple alternating-series reasoning.  \n\nThese layers of number–theoretic, analytic, and special-function techniques make the\nenhanced variant substantially more challenging than both the original problem and the\ncurrent kernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "calculation"
}