ai＞0，求 [a1+√(a1a2)+(a1a2a3)^(1/3)+…+(a1a2…an)^(1/n)]／(a1+a2+…+an) 最大值

王守恩

\(\frac{a_{1}}{a_{1}}=1.00000\)

\(\frac{a_{1}+\sqrt{a_{1}a_{2}}}{a_{1}+a_{2}}=1.20711\)

\(\frac{a_{1}+\sqrt{a_{1}a_{2}}+\sqrt[3]{a_{1}a_{2}a_{3}}}{a_{1}+a_{2}+a_{3}}=1.33333\)

\(\frac{a_{1}+\sqrt{a_{1}a_{2}}+\sqrt[3]{a_{1}a_{2}a_{3}}+\sqrt[4]{a_{1}a_{2}a_{3}a_{4}}}{a_{1}+a_{2}+a_{3}+a_{4}}=1.42084\)

\(\frac{a_{1}+\sqrt{a_{1}a_{2}}+\sqrt[3]{a_{1}a_{2}a_{3}}+\sqrt[4]{a_{1}a_{2}a_{3}a_{4}}+\sqrt[5]{a_{1}a_{2}a_{3}a_{4}a_{5}}}{a_{1}+a_{2}+a_{3}+a_{4}+a_{5}}= 1.48635\)

\(\frac{a_{1}+\sqrt{a_{1}a_{2}}+\sqrt[3]{a_{1}a_{2}a_{3}}+\sqrt[4]{a_{1}a_{2}a_{3}a_{4}}+\sqrt[5]{a_{1}a_{2}a_{3}a_{4}a_{5}}+\cdots}{a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+\cdots}=?\)

王守恩

就是不知道？？= ？？？——好玩！肯定是一个好玩的数字！！好玩！！！

{02, 1.20711}, {03, 1.33333}, {04, 1.42084}, {05, 1.48635}, {06, 1.53794}, {07, 1.58004}, {08, 1.61532}, {09, 1.64551}, {10, 1.67176}, {11, 1.69489}, {12, 1.71550},
{13, 1.73403}, {14, 1.75083}, {15, 1.76616}, {16, 1.78023}, {17, 1.79321}, {18, 1.80525}, {19, 1.81645}, {20, 1.82692}, {21, 1.83674}, {22, 1.84597}, {23, 1.85468},
{24, 1.86291}, {25, 1.87070}, {26, 1.87811}, {27, 1.88515}, {28, 1.89187}, {29, 1.89828}, {30, 1.90441}, {31, 1.91029}, {32, 1.91592}, {33, 1.92133}, {34, 1.92653},
{35, 1.93154}, {36, 1.93636}, {37, 1.94102}, {38, 1.94552}, {39, 1.94986}, {40, 1.95406}, {41, 1.95813}, {42, 1.96208}, {43, 1.96590}, {44, 1.96961}, {45, 1.97321},
{46, 1.97671}, {47, 1.98012}, {48, 1.98343}, {49, 1.98665}, {50, 1.98979}, {51, 1.99285}, {52, 1.99584}, {53, 1.99874}, {54, 2.00158}, {55, 2.00436}, {56, 2.00707},
{57, 2.00971}, {58, 2.01230}, {59, 2.01483}, {60, 2.01731}, {61, 2.01973}, {62, 2.02210}, {63, 2.02442}, {64, 2.02670}, {65, 2.02893}, {66, 2.03112}, {67, 2.03326},
{68, 2.03537}, {69, 2.03743}, {70, 2.03946}, {71, 2.04145}, {72, 2.04340}, {73, 2.04532}, {74, 2.04721}, {75, 2.04906}, {76, 2.05088}, {77, 2.05267}, {78, 2.05443},
{79, 2.05616}, {80, 2.05787}, {81, 2.05955}, {82, 2.06120}, {83, 2.06282}, {84, 2.06442}, {85, 2.06600}, {86, 2.06754}, {87, 2.06908}, {88, 2.07058}, {89, 2.07207},
{90, 2.07353}, {91, 2.07497}, {92, 2.07640}, {93, 2.07780}, {94, 2.07918}, {95, 2.08055}, {96, 2.08189}, {97, 2.08322}, {98, 2.08453}, {99, 2.08583}, {100,2.08711}}

王守恩

\(\frac{a_{1}+\sqrt{a_{1}a_{2}}+\sqrt[3]{a_{1}a_{2}a_{3}}+\sqrt[4]{a_{1}a_{2}a_{3}a_{4}}+\sqrt[5]{a_{1}a_{2}a_{3}a_{4}a_{5}}+\cdots}{a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+\cdots}=?\)

就是不知道？= ？？——肯定是一个好玩的数字！好玩！！我瞎掰1个！！！—— E - EulerGamma = 2.141066164

我们总可以认定 a1 = 1。特别地 n = 3,  可以有

Maximize[{(1 + Sqrt[a2] + Power[a2*a3, (3)^-1])/(1 + a2 + a3), a2 > a3 > 0}, {a2, a3}]

{4/3, {a2 -> 1/4, a3 -> 1/16}}

观察可知,  a1 > a2 > a3 > a4 > a5 > a6 > a7 > ......

记ak = P*a(k-i),  则P的分布呈现两头大中间小,  最小P出现在n/2。——蛮有规律。

譬如   n = 17   最大值 = 1.79321。P在 k = 9 值为 1.33819......

{"01", "1.00000000000000000`", "2.2606993333164427`"},
{"02", "0.44234099831975504`", "1.7218745364908050`"},
{"03", "0.25689502280534900`", "1.5395981147858715`"},
{"04", "0.16685849400450722`", "1.4493716292820742`"},
{"05", "0.11512471379556272`", "1.3975605188735323`"},
{"06", "0.08237547658283595`", "1.3662244409003148`"},
{"07", "0.06029424896582303`", "1.3478226567534772`"},
{"08", "0.04473455663006495`", "1.3389357696323970`"},
{"09", "0.03341053218881946`", "1.3381894675737530`"},
{"10", "0.02496696693435758`", "1.3456461207062560`"},
{"11", "0.01855388764562691`", "1.3627825778720222`"},
{"12", "0.01361470857266073`", "1.3932391022008292`"},
{"13", "0.00977198282129339`", "1.4450926941602615`"},
{"14", "0.00676218408741721`", "1.5377381951827709`"},
{"15", "0.00439748723716489`", "1.7299323551015682`"},
{"16", "0.00254199953206072`", "2.3154506472408736`"},
{"17", "0.00109784224297322`", },

最后说句心里话:  电脑还是玩不过人类的！！！

luyuanhong

王守恩

luyuanhong 发表于 2026-6-24 09:45

陆老师！这个——2.141066164——是怎么样的一个数？

\(\frac{a_{1}+\sqrt{a_{1}a_{2}}+\sqrt[3]{a_{1}a_{2}a_{3}}+\sqrt[4]{a_{1}a_{2}a_{3}a_{4}}+\sqrt[5]{a_{1}a_{2}a_{3}a_{4}a_{5}}+\cdots}{a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+\cdots}=?\)

就是不知道？= ？？——肯定是一个好玩的数字！好玩！！我瞎掰1个！！！—— E - EulerGamma = 2.141066164——

luyuanhong

luyuanhong

王守恩

luyuanhong 发表于 2026-6-26 01:07

就是不知道？= ？？——肯定是一个好玩的数字！好玩！！我瞎掰1个！！！—— E - EulerGamma = 2.141066164——就我的认知——人类还没有这个数。

我们总可以认定 a1 = 1。

观察可知,  a1 > a2 > a3 > a4 > a5 > a6 > a7 > ......

记ak = P*a(k-1),  当P是一个固定不变数，则最大值只能=2。

若要最大值>2。P的分布呈现两头大中间小,  最小P出现在n/2。——蛮有规律。

譬如——n = 57 最大值 = 2.00971......

最小 P 在 k = 29  值为 1.09624......

{01, 1.0000000, 1.99038},
{02, 0.5024160, 1.54083},
{03, 0.3260690, 1.38567},
{04, 0.2353150, 1.30589},
{05, 0.1801940, 1.25693},
{06, 0.1433600, 1.22370},
{07, 0.1171540, 1.19961},
{08, 0.0976594, 1.18135},
{09, 0.0826674, 1.16704},
{10, 0.0708352, 1.15553},
{11, 0.0613008, 1.14610},
{12, 0.0534863, 1.13825},
{13, 0.0469898, 1.13164},
{14, 0.0415235, 1.12602},
{15, 0.0368764, 1.12120},
{16, 0.0328901, 1.11705},
{17, 0.0294436, 1.11347},
{18, 0.0264432, 1.11036},
{19, 0.0238149, 1.10767},
{20, 0.0214999, 1.10535},
{21, 0.0194508, 1.10335},
{22, 0.0176289, 1.10164},
{23, 0.0160024, 1.10020},
{24, 0.0145450, 1.09900},
{25, 0.0132347, 1.09803},
{26, 0.0120532, 1.09728},
{27, 0.0109846, 1.09673},
{28, 0.0100158, 1.09639},
{29, 0.00913523, 1.09624},
{30, 0.00833321, 1.09630},
{31, 0.00760124, 1.09655},
{32, 0.00693195, 1.09701},
{33, 0.00631893, 1.09769},
{34, 0.00575657, 1.09860},
{35, 0.00523993, 1.09974},
{36, 0.00476468, 1.10116},
{37, 0.00432697, 1.10286},
{38, 0.00392339, 1.10489},
{39, 0.00355093, 1.10729},
{40, 0.00320686, 1.11011},
{41, 0.00288879, 1.11341},
{42, 0.00259455, 1.11728},
{43, 0.00232220, 1.12184},
{44, 0.00206999, 1.12723},
{45, 0.00183636, 1.13364},
{46, 0.00161988, 1.14134},
{47, 0.00141928, 1.15071},
{48, 0.00123339, 1.16229},
{49, 0.00106117, 1.17689},
{50, 0.00090167, 1.19577},
{51, 0.00075405, 1.22106},
{52, 0.000617535, 1.25657},
{53, 0.000491443, 1.30991},
{54, 0.000375174, 1.39879},
{55, 0.000268213, 1.57633},
{56, 0.000170151, 2.10778},
{57, 0.0000807248, ""}

王守恩

这个代码应该没问题——就是往前走不了。

1. Table[V = Table[Subscript[A, i], {i, n}]; {n, NMaximize[{Sum[Power[Product[V[[j]], {j, k}], 1/k], {k, n}]/Total[V],
   
2. Thread[V > 0]}, V, WorkingPrecision -> 20, MaxIterations -> 20 n, Method -> "DifferentialEvolution"][[1]]}, {n, 30}]

复制代码

{01, 1.0000000000000000000}, {02, 1.2071067811865475244}, {03, 1.3333333333333333333}, {04, 1.4208443854096138127}, {05, 1.4863532289630506405},
{06, 1.5379375565200349314}, {07, 1.5800372106320523521}, {08, 1.6153223997025154146}, {09, 1.6455095229654850634}, {10, 1.6717598117746269016},
{11, 1.6948914447952002823}, {12, 1.7155002229834597560}, {13, 1.7340320305579521193}, {14, 1.7507752355237590930}, {15, 1.7661557795845829301},
{16, 1.7802262265671340380}, {17, 1.7932104410890079583}, {18, 1.8052479032405679001}, {19, 1.8164538550045598629}, {20, 1.8269244980919275537},
{21, 1.8367408861507595618}, {22, 1.8459718828800575196}, {23, 1.8546764405356997440}, {24, 1.8629053764142772740}, {25, 1.8707027733414674741},
{26, 1.8781070949833550147}, {27, 1.8851520823506100042}, {28, 1.8918674806263563208}, {29, 1.8982796331206052650}, {30, 1.9044119702229642481}}

王守恩

\(\frac{a_{1}+\sqrt{a_{1}a_{2}}+\sqrt[3]{a_{1}a_{2}a_{3}}+\sqrt[4]{a_{1}a_{2}a_{3}a_{4}}+\sqrt[5]{a_{1}a_{2}a_{3}a_{4}a_{5}}+\cdots}{a_{1}+a_{2}+a_{3}+a_{4}+a_{5}+\cdots}=?\)

{001, 1.00000}, {002, 1.20711}, {003, 1.33333}, {004, 1.42084}, {005, 1.48635}, {006, 1.53794}, {007, 1.58004}, {008, 1.61532}, {009, 1.64551}, {010, 1.67176},
{011, 1.69489}, {012, 1.71550}, {013, 1.73403}, {014, 1.75083}, {015, 1.76616}, {016, 1.78023}, {017, 1.79321}, {018, 1.80525}, {019, 1.81645}, {020, 1.82692},
{021, 1.83674}, {022, 1.84597}, {023, 1.85468}, {024, 1.86291}, {025, 1.87070}, {026, 1.87811}, {027, 1.88515}, {028, 1.89187}, {029, 1.89828}, {030, 1.90441},
{031, 1.91029}, {032, 1.91592}, {033, 1.92133}, {034, 1.92653}, {035, 1.93154}, {036, 1.93636}, {037, 1.94102}, {038, 1.94552}, {039, 1.94986}, {040, 1.95406},
{041, 1.95813}, {042, 1.96208}, {043, 1.96590}, {044, 1.96961}, {045, 1.97321}, {046, 1.97671}, {047, 1.98012}, {048, 1.98343}, {049, 1.98665}, {050, 1.98979},
{051, 1.99285}, {052, 1.99584}, {053, 1.99874}, {054, 2.00158}, {055, 2.00436}, {056, 2.00707}, {057, 2.00971}, {058, 2.01230}, {059, 2.01483}, {060, 2.01731},
{061, 2.01973}, {062, 2.02210}, {063, 2.02442}, {064, 2.02670}, {065, 2.02893}, {066, 2.03112}, {067, 2.03326}, {068, 2.03537}, {069, 2.03743}, {070, 2.03946},
{071, 2.04145}, {072, 2.04340}, {073, 2.04532}, {074, 2.04721}, {075, 2.04906}, {076, 2.05088}, {077, 2.05267}, {078, 2.05443}, {079, 2.05616}, {080, 2.05787},
{081, 2.05955}, {082, 2.06120}, {083, 2.06282}, {084, 2.06442}, {085, 2.06600}, {086, 2.06754}, {087, 2.06908}, {088, 2.07058}, {089, 2.07207}, {090, 2.07353},
{091, 2.07497}, {092, 2.07640}, {093, 2.07780}, {094, 2.07918}, {095, 2.08055}, {096, 2.08189}, {097, 2.08322}, {098, 2.08453}, {099, 2.08583}, {100, 2.08711},
{101, 2.08837}, {102, 2.08961}, {103, 2.09084}, {104, 2.09206}, {105, 2.09325}, {106, 2.09444}, {107, 2.09561}, {108, 2.09677}, {109, 2.09791}, {110, 2.09904},
{111, 2.10016}, {112, 2.10126}, {113, 2.10235}, {114, 2.10343}, {115, 2.10450}, {116, 2.10555}, {117, 2.10660}, {118, 2.10763}, {119, 2.10865}, {120, 2.10966},
{121, 2.11066}, {122, 2.11165}, {123, 2.11262}, {124, 2.11359}, {125, 2.11455}, {126, 2.11550}, {127, 2.11644}, {128, 2.11737}, {129, 2.11829}, {130, 2.11920},
{131, 2.12010}, {132, 2.12145}, {133, 2.12188}, {134, 2.12276}, {135, 2.12363}, {136, 2.12449}, {137, 2.12534}, {138, 2.12618}, {139, 2.12702}, {140, 2.12785},
{141, 2.12867}, {142, 2.12948}, {143, 2.13029}, {144, 2.13109}, {145, 2.13188}, {146, 2.13267}, {147, 2.13344}, {148, 2.13422}, {149, 2.13498}, {150, 2.13574},
{151, 2.13649}, {152, 2.13724}, {153, 2.13798}, {154, 2.13871}, {155, 2.13944}, {156, 2.14016}, {157, 2.14087}, {158, 2.14158}, {159, 2.14229}, {160, 2.14299},
{161, 2.14368}, {162, 2.14437}, {163, 2.14505}, {164, 2.14572}, {165, 2.14640}, {166, 2.14707}, {167, 2.14772}, {168, 2.14838}, {169, 2.14903}, {170, 2.14968},
{171, 2.15032}, {172, 2.15095}, {173, 2.15159}, {174, 2.15221}, {175, 2.15283}, {176, 2.15346}, {177, 2.15407}, {178, 2.15468}, {179, 2.15528}, {180, 2.15588}}