单位分数求和

王守恩

单位分数求和
1，求证。下面所有单位分数的和小于1。
    1/2^3+1/3^3+1/4^3+1/5^3+1/6^3.........
    1/3^3+1/4^3+1/5^3+1/6^3+1/7^3.........
    1/4^3+1/5^3+1/6^3+1/7^3+1/8^3.........  
    1/5^3+1/6^3+1/7^3+1/8^3+1/9^3.........  
    1/6^3+1/7^3+1/8^3+1/9^3+1/10^3.......
    .......................


2，求证。下面所有单位分数的和是无穷大。
    1/3^2+1/4^2+1/5^2+1/6^2+1/7^2.........
    1/4^2+1/5^2+1/6^2+1/7^2+1/8^2.........  
    1/5^2+1/6^2+1/7^2+1/8^2+1/9^2.........  
    1/6^2+1/7^2+1/8^2+1/9^2+1/10^2.......  
    1/7^2+1/8^2+1/9^2+1/10^2+1/11^2.....
    .......................

manthanein

https://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constant

hujunhua

第1题, 分3步
1）如果你不想使用定积分的话，可以证明一个裂项不等式`\frac1{(n+1)^3}<\frac1{2n^2}-\frac1{2(n+1)^2}`
2）第 i 行裂项相消得到其和`<\frac1{2i^2}`，并再次裂项`\frac1{2i^2}<\frac1{2(i-1)}-\frac1{2i}`
3）所以各行的结果再相加相消的结果就是小于1.

第2题，仿上先证明一个裂项不等式`\frac1{(n+2)^2}>\frac1{n+2}-\frac1{n+3}`, 然后第 i 行裂项相消得其和`>\frac1{i+2}`, 于是所有行按顺序是一个调和数列，其和发散。

northwolves

\[ \lim_{n \to \infty} \sum_{k=3}^n \frac{1}{k^2} = \frac{\pi^2}{6}-\frac{5}{4}\]

王守恩

单位分数求和
1，求证。下面所有单位分数的和小于1。
    1/2^3+1/3^3+1/4^3+1/5^3+1/6^3.........
    1/3^3+1/4^3+1/5^3+1/6^3+1/7^3.........
    1/4^3+1/5^3+1/6^3+1/7^3+1/8^3.........  
    1/5^3+1/6^3+1/7^3+1/8^3+1/9^3.........  
    1/6^3+1/7^3+1/8^3+1/9^3+1/10^3.......
    .......................


2，求证。下面所有单位分数的和是无穷大。
  
    1/2^2+1/3^2+1/4^2+1/5^2+1/6^2.........
    1/3^2+1/4^2+1/5^2+1/6^2+1/7^2.........
    1/4^2+1/5^2+1/6^2+1/7^2+1/8^2.........  
    1/5^2+1/6^2+1/7^2+1/8^2+1/9^2.........  
    1/6^2+1/7^2+1/8^2+1/9^2+1/10^2.......
    .......................


3，问：当n=?时，下面所有单位分数的和等于1。
    1/2^n+1/3^n+1/4^n+1/5^n+1/6^n.........
    1/3^n+1/4^n+1/5^n+1/6^n+1/7^n.........
    1/4^n+1/5^n+1/6^n+1/7^n+1/8^n.........  
    1/5^n+1/6^n+1/7^n+1/8^n+1/9^n.........  
    1/6^n+1/7^n+1/8^n+1/9^n+1/10^n.......
    .......................

王守恩

\(\D\frac{1}{2}=\frac{\frac{1}{1+2521}+\frac{1}{2+2522}+\frac{1}{3+2523}+\frac{1}{4+2524}+\cdots+\frac{1}{1260+3780}}{1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\cdots+\frac{1}{2519}-\frac{1}{2520}}\)

\(\D\frac{1}{2}=\frac{\frac{1}{1+1681}+\frac{1}{2+1682}+\frac{1}{3+1683}+\frac{1}{4+1684}+\cdots+\frac{1}{1680+3360}}{1+\frac{1}{2}-\frac{2}{3}+\frac{1}{4}+\frac{1}{5}-\frac{2}{6}+\cdots+\frac{1}{2518}+\frac{1}{2519}-\frac{2}{2520}}\)

\(\D\frac{1}{2}=\frac{\frac{1}{1+1261}+\frac{1}{2+1262}+\frac{1}{3+1263}+\frac{1}{4+1264}+\cdots+\frac{1}{1890+3150}}{1+\frac{1}{2}+\frac{1}{3}-\frac{3}{4}+\cdots+\frac{1}{2517}+\frac{1}{2518}+\frac{1}{2519}-\frac{3}{2520}}\)

\(\D\frac{1}{2}=\frac{\frac{1}{1+1009}+\frac{1}{2+1010}+\frac{1}{3+1011}+\frac{1}{4+1012}+\cdots+\frac{1}{2016+3024}}{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{4}{5}+\cdots+\frac{1}{2516}+\frac{1}{2517}+\frac{1}{2518}+\frac{1}{2519}-\frac{4}{2520}}\)

\(\D\frac{1}{2}=\frac{\frac{1}{1+841}+\frac{1}{2+842}+\frac{1}{3+843}+\frac{1}{4+844}+\cdots+\frac{1}{2100+2940}}{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{5}{6}+\cdots+\frac{1}{2515}+\frac{1}{2516}+\frac{1}{2517}+\frac{1}{2518}+\frac{1}{2519}-\frac{5}{2520}}\)

\(\D\frac{1}{2}=\frac{\frac{1}{1+721}+\frac{1}{2+722}+\frac{1}{3+723}+\frac{1}{4+724}+\cdots+\frac{1}{2160+2880}}{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\frac{6}{7}+\cdots+\frac{1}{2514}+\frac{1}{2515}+\frac{1}{2516}+\frac{1}{2517}+\frac{1}{2518}+\frac{1}{2519}-\frac{6}{2520}}\)

\(\D\frac{1}{2}=\frac{\frac{1}{1+631}+\frac{1}{2+632}+\frac{1}{3+633}+\frac{1}{4+634}+\cdots+\frac{1}{2205+2835}}{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}-\frac{7}{8}+\cdots+\frac{1}{2013}+\frac{1}{2014}+\frac{1}{2015}+\frac{1}{2016}+\frac{1}{2017}+\frac{1}{2018}+\frac{1}{2019}-\frac{7}{2520}}\)

\(\D\frac{1}{2}=\frac{\frac{1}{1+561}+\frac{1}{2+562}+\frac{1}{3+563}+\frac{1}{4+564}+\cdots+\frac{1}{2240+2800}}{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}-\frac{8}{9}+\cdots+\frac{1}{2012}+\frac{1}{2013}+\frac{1}{2014}+\frac{1}{2015}+\frac{1}{2016}+\frac{1}{2017}+\frac{1}{2018}+\frac{1}{2019}-\frac{8}{2520}}\)

\(\D\frac{1}{2}=\frac{\frac{1}{1+501}+\frac{1}{2+502}+\frac{1}{3+503}+\frac{1}{4+504}+\cdots+\frac{1}{2270+2770}}{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}-\frac{9}{10}+\cdots+\frac{1}{2011}+\frac{1}{2012}+\frac{1}{2013}+\frac{1}{2014}+\frac{1}{2015}+\frac{1}{2016}+\frac{1}{2017}+\frac{1}{2018}+\frac{1}{2019}-\frac{9}{2520}}\)

lihpb00

看不懂问题。
    1/3^2+1/4^2+1/5^2+1/6^2+1/7^2.........
这个级数是收敛的

王守恩

S(A)=1+1/2+1/3+1/4+1/5+1/6+...+1/A.

S(A) > ln(A) + 1/(2A) - 1/(12A^2+1) + r.

S(A) < ln(A) + 1/(2A) - 1/(12A^2+2) + r.

其中: r=0.57721566490153286061...

{2.9289682540, 5.1873775176396203, 7.4854708605503449126565, 9.7876060360443822641784779049, 12.090146129863427947363219363504220, 14.392726722865723631381127493188587676645,
16.695311365859851815399118939540451884249869752, 18.997896413853898324417110394223982841850971244970103}      前面8个数=准确值。
{2.9289682571, 5.1873775176396234, 7.4854708605503449126597, 9.7876060360443822641784779080, 12.090146129863427947363219363504223, 14.392726722865723631381127493188587676648,
16.695311365859851815399118939540451884249869756, 18.997896413853898324417110394223982841850971244970107, 21.300481502347944016685101848908346966127072733598880383105,
23.603066594891989700785593303592711173719841722227653359139046244, 25.905651687841035384804409758277075381320942377606426335172374145528166,
28.208236780830581068822409462961439588922043866151874311205702046495739042905, 30.510821873824176752840401000145803796523145354780638954739029947463311652583246913,
32.813406966818177436858392455655168004124246843409411929939107848430884262260599394240907, 35.115992059812218620876383910347782211725348332038184905972352424398456871937951874476905139635,
37.418577152806263854894375365032228919326449820666957882005680317033529481615304354712902344725319522,
39.721162245800309493912366819716593951927551309295730858039008218000268841292656834948899549814917821328083,
42.023747338794355173430358274400958167778652797924503834072336118967841367645009315184896754904516119670051560010,
44.326332431788400856998349729085322375462254286553276810105664019935413977314029295420893959994114418012019344053245806,
46.628917524782446540971341183769686583064180775182049786138991920902986586991380942406891165083712716353987128095532055149150,
48.931502617776492224984832638454050790665290513810822762172319821870559196668733422559563370173311014695954912137818303782560538780,
51.234087710770537909002374093138414998266392084939595738205647722838131806346085902795552242762909313037922696180104552415970064246308291,
53.536672803764583593020320547822779205867493574393368714238975623805704416023438383031549447019257611379890480222390801049379589711391098239548,
55.839257896758629277038307502507143413468595063030391690272303524773277025700790863267546652108772584721858264264677049682789115176473904996215070855,
58.141842989752674961056298507191507621069696551659247166305631425740849635378143343503543857198370874731326048306963298316198640641556711752881358225114465,
60.444428082746720645074289916875871828670798040288020967338959326708422245055495823739541062287969173072460582349249546949608166106639518509547645594213247462808,
62.747013175740766329092281367060236036271899528916793951622287227675994854732848303975538267377567471414428283066535795583017691571722325266213932963312029152291994958,
65.049598268734812013110272821294600243873001017545566927738115128643567464410200784211535472467165769756396067100489544216427217036805132022880220332410810841774902166858468,
67.352183361728857697128264275933964451474102506174339903772268029611140074087553264447532677556764068098363851142774959599836742501887938779546507701509592531257809375184023082207,
69.654768454722903381146255730613828659075203994803112879805604180578712683764905744683529882646362366440331635185061208149921267966970745536212795070608374220740716583509577763051670787,
71.957353547716949065164247185297742866676305483431885855838932164046285293442258224919527087735960664782299419227347456783322460932053552292879082439707155910223623791835132443895470682405824,
74.259938640710994749182238639982062074277406972060658831872260065838857903119610705155524292825558963124267203269633705416731985563886359049545369808805937599706531000160687124739270577232057611155,
76.56252373370504043320023009466642178187850846068943180790558796681468051279696318539152149791515726146623498731191995405014151102888584080621165717790471928918943820848624180558307047205829080968408059
78.86510882669908611721822154935078553947960994931820478393891586778233562247431566562751870300475555980820277135420620268355103649396863923\03779445470035009786723454168117964864268703668845240082124740
81.16769391969313180123621300403514970208071143794697775997224376874990905715166814586351590809435385815017055539649245131696056195905144598621098191610228266815525262513735116727067026171075720674086760
83.47027901268717748525420445871951390518181292657575073600557166971748167507902062609951311318395215649213833943877869995037008742413425274287718596020106435763815983346290584811447015653699040526926110

前面8个数=准确值。要把第9个, 第10个, 第11个准确值搞出来就难了。

后面的数利用简单公式就可以出来!    ln(A) + 1/(2A) - 5/(60A^2+6) + r,

当 A = 10时, 8位小数有效。

当 A = 100时, 14位小数有效。

当 A = 1000时, 20位小数有效。

当 A = 10000时, 26位小数有效。

当 A = 100000时, 32位小数有效。
......

1. Table[N[Log[10^A] + 1/(2*10^A) - 5/(60*10^(2 A) + 6) + EulerGamma, 6 A + 5], {A, 99}]

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