各位大侠(f+1)1+(f+2)2+(f+3)3+...+(f+2008)2008=(f+2009)2009.求f

wayne

将楼主的2008改成 一般性的n，谁能给出 方程的实根个数的表达式 ？
$(x+1)^1 + (x+2)^2 + (x+3)^3 + ... + (x+n)^n= (x+n+1)^{n+1}$
我算了一下 n<=500的所有情况

1. Table[{n,CountRoots[Sum[(x+i)^i,{i,n}]-(x+n+1)^(n+1),x]},{n,500}]

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1. {1, 0}, {2, 1}, {3, 2}, {4, 1}, {5, 2}, {6, 3}, {7, 4}, {8, 3}, {9, 4}, {10, 3},
   
2. {11, 2}, {12, 3}, {13, 4}, {14, 3}, {15, 4}, {16, 5}, {17, 4}, {18, 5}, {19, 6}, {20, 5},
   
3. {21, 6}, {22, 5}, {23, 6}, {24, 7}, {25, 6}, {26, 7}, {27, 8}, {28, 7}, {29, 8}, {30, 7},
   
4. {31, 6}, {32, 7}, {33, 8}, {34, 7}, {35, 8}, {36, 7}, {37, 8}, {38, 9}, {39, 8}, {40, 9},
   
5. {41, 10}, {42, 9}, {43, 10}, {44, 9}, {45, 10}, {46, 11}, {47, 10}, {48, 11}, {49, 12}, {50, 11},
   
6. {51, 12}, {52, 9}, {53, 10}, {54, 11}, {55, 10}, {56, 11}, {57, 12}, {58, 11}, {59, 12}, {60, 11},
   
7. {61, 12}, {62, 13}, {63, 12}, {64, 13}, {65, 14}, {66, 13}, {67, 14}, {68, 13}, {69, 14}, {70, 15},
   
8. {71, 14}, {72, 15}, {73, 16}, {74, 15}, {75, 16}, {76, 15}, {77, 16}, {78, 15}, {79, 14}, {80, 15},
   
9. {81, 16}, {82, 15}, {83, 16}, {84, 15}, {85, 16}, {86, 17}, {87, 16}, {88, 17}, {89, 16}, {90, 17},
   
10. {91, 18}, {92, 17}, {93, 18}, {94, 19}, {95, 18}, {96, 19}, {97, 18}, {98, 19}, {99, 20}, {100, 19},
    
11. {101, 20}, {102, 19}, {103, 20}, {104, 21}, {105, 20}, {106, 19}, {107, 20}, {108, 19}, {109, 20}, {110, 19},
    
12. {111, 20}, {112, 21}, {113, 20}, {114, 21}, {115, 20}, {116, 21}, {117, 22}, {118, 21}, {119, 22}, {120, 23},
    
13. {121, 22}, {122, 23}, {123, 22}, {124, 23}, {125, 24}, {126, 23}, {127, 24}, {128, 23}, {129, 24}, {130, 25},
    
14. {131, 24}, {132, 25}, {133, 24}, {134, 25}, {135, 24}, {136, 23}, {137, 24}, {138, 25}, {139, 24}, {140, 25},
    
15. {141, 24}, {142, 25}, {143, 26}, {144, 25}, {145, 26}, {146, 25}, {147, 26}, {148, 27}, {149, 26}, {150, 27},
    
16. {151, 26}, {152, 27}, {153, 28}, {154, 27}, {155, 28}, {156, 29}, {157, 28}, {158, 29}, {159, 28}, {160, 29},
    
17. {161, 30}, {162, 29}, {163, 30}, {164, 29}, {165, 30}, {166, 29}, {167, 28}, {168, 29}, {169, 28}, {170, 29},
    
18. {171, 30}, {172, 29}, {173, 30}, {174, 29}, {175, 30}, {176, 31}, {177, 30}, {178, 31}, {179, 30}, {180, 31},
    
19. {181, 32}, {182, 31}, {183, 32}, {184, 33}, {185, 32}, {186, 33}, {187, 32}, {188, 33}, {189, 34}, {190, 33},
    
20. {191, 34}, {192, 33}, {193, 34}, {194, 35}, {195, 34}, {196, 35}, {197, 34}, {198, 33}, {199, 34}, {200, 33},
    
21. {201, 34}, {202, 33}, {203, 34}, {204, 35}, {205, 34}, {206, 35}, {207, 34}, {208, 35}, {209, 36}, {210, 35},
    
22. {211, 36}, {212, 35}, {213, 36}, {214, 37}, {215, 36}, {216, 37}, {217, 36}, {218, 37}, {219, 38}, {220, 37},
    
23. {221, 38}, {222, 37}, {223, 38}, {224, 39}, {225, 38}, {226, 39}, {227, 40}, {228, 39}, {229, 40}, {230, 39},
    
24. {231, 38}, {232, 39}, {233, 38}, {234, 39}, {235, 38}, {236, 39}, {237, 40}, {238, 39}, {239, 40}, {240, 39},
    
25. {241, 40}, {242, 41}, {243, 40}, {244, 41}, {245, 40}, {246, 41}, {247, 42}, {248, 41}, {249, 42}, {250, 41},
    
26. {251, 42}, {252, 43}, {253, 42}, {254, 43}, {255, 42}, {256, 43}, {257, 44}, {258, 43}, {259, 44}, {260, 43},
    
27. {261, 44}, {262, 45}, {263, 44}, {264, 45}, {265, 44}, {266, 43}, {267, 44}, {268, 43}, {269, 44}, {270, 43},
    
28. {271, 44}, {272, 45}, {273, 44}, {274, 45}, {275, 44}, {276, 45}, {277, 46}, {278, 45}, {279, 46}, {280, 45},
    
29. {281, 46}, {282, 47}, {283, 46}, {284, 47}, {285, 46}, {286, 47}, {287, 48}, {288, 47}, {289, 48}, {290, 47},
    
30. {291, 48}, {292, 49}, {293, 48}, {294, 49}, {295, 48}, {296, 49}, {297, 50}, {298, 49}, {299, 50}, {300, 49},
    
31. {301, 48}, {302, 49}, {303, 48}, {304, 49}, {305, 48}, {306, 49}, {307, 50}, {308, 49}, {309, 50}, {310, 49},
    
32. {311, 50}, {312, 51}, {313, 50}, {314, 51}, {315, 50}, {316, 51}, {317, 52}, {318, 51}, {319, 52}, {320, 51},
    
33. {321, 52}, {322, 53}, {323, 52}, {324, 53}, {325, 52}, {326, 53}, {327, 54}, {328, 53}, {329, 54}, {330, 53},
    
34. {331, 54}, {332, 55}, {333, 54}, {334, 55}, {335, 54}, {336, 55}, {337, 56}, {338, 53}, {339, 54}, {340, 53},
    
35. {341, 54}, {342, 55}, {343, 54}, {344, 55}, {345, 54}, {346, 55}, {347, 56}, {348, 55}, {349, 56}, {350, 55},
    
36. {351, 56}, {352, 57}, {353, 56}, {354, 57}, {355, 56}, {356, 57}, {357, 58}, {358, 57}, {359, 58}, {360, 57},
    
37. {361, 58}, {362, 57}, {363, 58}, {364, 59}, {365, 58}, {366, 59}, {367, 58}, {368, 59}, {369, 60}, {370, 59},
    
38. {371, 60}, {372, 59}, {373, 60}, {374, 61}, {375, 58}, {376, 59}, {377, 58}, {378, 59}, {379, 60}, {380, 59},
    
39. {381, 60}, {382, 59}, {383, 60}, {384, 61}, {385, 60}, {386, 61}, {387, 60}, {388, 61}, {389, 62}, {390, 61},
    
40. {391, 62}, {392, 61}, {393, 62}, {394, 63}, {395, 62}, {396, 63}, {397, 62}, {398, 63}, {399, 64}, {400, 63},
    
41. {401, 64}, {402, 63}, {403, 64}, {404, 65}, {405, 64}, {406, 65}, {407, 64}, {408, 65}, {409, 66}, {410, 65},
    
42. {411, 66}, {412, 65}, {413, 64}, {414, 65}, {415, 64}, {416, 65}, {417, 64}, {418, 65}, {419, 64}, {420, 65},
    
43. {421, 66}, {422, 65}, {423, 66}, {424, 65}, {425, 66}, {426, 67}, {427, 66}, {428, 67}, {429, 66}, {430, 67},
    
44. {431, 68}, {432, 67}, {433, 68}, {434, 67}, {435, 68}, {436, 69}, {437, 68}, {438, 69}, {439, 68}, {440, 69},
    
45. {441, 70}, {442, 69}, {443, 70}, {444, 69}, {445, 70}, {446, 71}, {447, 70}, {448, 71}, {449, 70}, {450, 71},
    
46. {451, 72}, {452, 69}, {453, 70}, {454, 69}, {455, 70}, {456, 71}, {457, 70}, {458, 71}, {459, 70}, {460, 71},
    
47. {461, 70}, {462, 71}, {463, 72}, {464, 71}, {465, 72}, {466, 71}, {467, 72}, {468, 73}, {469, 72}, {470, 73},
    
48. {471, 72}, {472, 73}, {473, 74}, {474, 73}, {475, 74}, {476, 73}, {477, 74}, {478, 75}, {479, 74}, {480, 75},
    
49. {481, 74}, {482, 75}, {483, 76}, {484, 75}, {485, 76}, {486, 75}, {487, 76}, {488, 77}, {489, 76}, {490, 77},
    
50. {491, 74}, {492, 75}, {493, 74}, {494, 75}, {495, 76}, {496, 75}, {497, 76}, {498, 75}, {499, 76}, {500, 77}

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wayne

mathe 发表于 2010-4-6 13:19
大概-1919.715，不过计算误差影响很大

不错，跟我上面给的一个根 -1919.7150681255258177 有效位基本一致。

@KeyTo9_Fans ,只是，跟 Fans的答案 差别就大了。

云梦

wayne 发表于 2013-8-24 18:23
我倒，花了2308.453125秒钟，算出有269个实根, 下面给的结果都是精确度最后一位，
谁有兴趣算一下，跟 ...

仔细检查一下程序，结果有问题。

wayne

云梦 发表于 2013-8-24 20:10
仔细检查一下程序，结果有问题。

问题在哪，可否指明，
我的程序很简单，先是计算n比较小的情况，可以得出高精度的表达式，容易验证。
所以，我就直接n=2008，让机器长时间跑了：

1. n=2008;Solve[Sum[(x+i)^i,{i,n}]==(x+n+1)^(n+1),x,Reals,WorkingPrecision->20]

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wayne

还可以用RootIntervals 函数确定多项式方程 所有实根的区间。
这个也能证实实根的个数

1. n = 500; RootIntervals[Sum[(x + i)^i, {i, n}] - (x + n + 1)^(n + 1)]

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或者CountRoots 函数，直接计算多项式方程 实根的个数

1. n = 500; CountRoots[Sum[(x + i)^i, {i, n}] - (x + n + 1)^(n + 1), x]

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云梦

Dim Ex As Chen
Dim Px As Chen
Dim hx As Chen
Dim Wx As Chen
Dim Pz As Chen
Dim i As Long
Dim Fz As String
Call SJbz
Ex = Ax

    Ax = Ex
    Bx.Bz = ""
    Bx.st = "2009"
    Bx.zs = 3
    Px = Bx
    hsfx = "+"
    Call Addition
    Call Lnx
    Cx = Px
    Call Mult_
    Ax = Cx
    Call Exp
    Pz = Ax

hx.Bz = ""
hx.st = "0"
hx.zs = 0
For i = 2008 To 1 Step -1
    Ax = Ex
    Bx.Bz = ""
    Bx.st = CStr(i)
    Bx.zs = Len(Bx.st) - 1
    Px = Bx
    hsfx = "+"
    Call Addition
    Fz = Ax.Bz
    Ax.Bz = ""
    Call Lnx
    Cx = Px
    Call Mult_
    Ax = Cx
    Call Exp
    Ax.Bz = IIf(i Mod 2 = 0, "", Fz)
    Wx = Ax
    Bx = Wx
    Ax = hx
    hsfx = "+"
    Call Addition
    hx = Ax
    If Wx.zs - Pz.zs < -xChen Then Exit For
Next

    Cx = Pz
    Ax = hx
    Call Multx_
    Ax = Cx
Call Sc(Ax.Bz, Ax.st, Ax.zs)

云梦

问题出现在n-k的变号.掌握这个特点，计算精度不是主要问题。

wayne

云梦 发表于 2013-8-24 12:45
-1979.1730271145406331002572745293671240009296844556718201337235736730502757699569793452441351230244 ...

不错，完全一致。该根接下来的数字是8599 。。。
================
既然 云梦给出了750位的数字字符，我也给到这个程度，269个实根，都精确到小数点后750位，(20楼才给出20位)：

2008.data (198.48 KB, 下载次数: 5)

2013-8-24 23:30 上传

点击文件名下载附件

1. -2223.581966210305147566369095549758856082460726047951486981929204551381908270008938095672195203118278905870747530295446231407784323995145298292152483444406572401094285295060724743341758768643575433116154027690150478479110510582111913719431839847721478492080472802902468948682483966095237949758631531793160551861988118668732745962066764703836521033301945334148528488666372155967936743490461380609066676906377376681364499952823199195650253868019733046017082520761482807559980975157053197614961259174691588968083085431625957251168357524264022504582026234072035641781397666454744905213604096068689486066476750136575994990054138089624134967261459528991855254759123610741825499889113604856508255581626735207924128223369062228786811605817818075105933955202207
   
2. -2222.448434521644097040711079773942612431347421024289020216344438682891586987415286880912500588609565077543306070296954092933899705477425288089114013662271385912470106450714846802643628710315175154578545259115591694401858502003419639119828881104941314618112220482041667521406183474737741665202255670918154109563135336838389133297145317959903468013973194298065824803385050986015302736263336821489704878784384150054977000706299368618342588373622550045339205014378682862033528486573237290500998430309020787300511235936111397773647558448096363352139959545406189421575692087984451068425780243568474191095434392648355888925360905054963684971397132129869432422818805569213298707000507975626910690113194032591422865126530106574390642927339513169530789850991872
   
3. -2221.307145233023912439381649245225552767139355064102032576284072934541355897287012331861344713465710372696122360956660733758041870378132651779770959735833542435527037856522963646602793871355503287584488475840950060618133326628052415720005586878795398689538166992166137655162250762990666434958201598090804533879857517932168543934064770454394929269980584799441773397403856135984923525262032673459858981478545755148809345699336406586853277777350091340002115341389792080391426037847652578182222467090192480057065356548478900136251482517597346385216567028356689566349639998127159837627287696020112239669098240621491260495850855800324900461060415455075180498456039895843409987710411466977045901648081925777028563731345214930300383178711996016749649849132958
   
4. -2220.165968792058961079953106087633414999657387959921897458528437920227238385528206815350855644842366149836169067642155789070702566637803369835267182165655384195031592116988966975724661143603434814270076688270514442935116193057553706128470747606761703447629417837770431901170725353520588657370594981456786538518347196765067064209304635243968569376414581418549930164654346279092753604548822201317663728129993166622068854925821368158694422135956186405017546474694662543557227980470020034562168489026011766672164528035811757561501662096994005135531501263085839772630369921750558483433984046326009213277602236495362949374928444784329149225236163257037970574778982559353361850661441664102093578263756254918512486298715823317686606108676901699745430860046552
   
5. -2219.024781876809047446685877541814972387157200113322465545191658585895011600333486801935652554532367336131097646616944705642081003389580862620086947861707775267823097079856400996879725623478787231290979481540269571640986079638785243638747744566720824842091491200475233061628435581052521787865604502186557685210558518267136389433340298473870887660661075145011574929135263324750431592872683936912906659975732407217129990635547959836903792845091986014427912698769576612399810598791413275892279471389860312164404181539976175307138268211650856223398211623682085776406551739590281304375023222652476272360755544213424990098678760104355090343892252278864694324420062027120342463445524931385097374860657101498660004038520726962898033308495415279560229774535243
   
6. .....
   
7. 详情见附件

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wayne

云梦 发表于 2013-8-24 21:20
问题出现在n-k的变号.掌握这个特点，计算精度不是主要问题。

额，要是让我放弃Mathematica,这题我还没法做呢。
我看了你的VB程序，Chen类型就是 你之前开发的那个高精度计算库里的高精度类型吗，
大致思路就是科学计算法表示浮点数？

云梦

是的，这道题的正确结果只有一个：
约等于-1918.0616280454
其实不如把a=b改成a/b=1更明确,可以保证不同精度的要求。
a/b=0.999999998931504681999802

(n+2009)^2009有3936位，按差值计算必须保证精度在4000位以上，改成按比例精度可以降到128位，甚至64位就可以了。