[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/19448#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/19448","headline":"\u30cf\u30fc\u30e2\u30cb\u30b7\u30e5\u30fb\u30ec\u30a4\u30d8 &#8211; \u30a6\u30a3\u30ad\u30da\u30c7\u30a3\u30a2","name":"\u30cf\u30fc\u30e2\u30cb\u30b7\u30e5\u30fb\u30ec\u30a4\u30d8 &#8211; \u30a6\u30a3\u30ad\u30da\u30c7\u30a3\u30a2","description":"before-content-x4 \u8abf\u548c\u306e\u3068\u308c\u305f\u30b7\u30ea\u30fc\u30ba \u624b\u8db3\u3092\u5408\u8a08\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u6570\u5b66\u306e\u30b7\u30ea\u30fc\u30ba\u3067\u3059 \u521d\u3081 \u3001 12\u3001 13\u3001 14\u3001 15\u3001 &#8230; {displaystyle 1\u3001{tfrac {1} {2}}\u3001{tfrac {1} {3}}\u3001{tfrac {1} {4}}\u3001{tfrac {1} {5}}\u3001dotc}}","datePublished":"2022-09-17","dateModified":"2022-09-17","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/44a4cee54c4c053e967fe3e7d054edd4?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c9e1b856256a964035ab28d979d6ac4ff257f584","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/c9e1b856256a964035ab28d979d6ac4ff257f584","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/all2jp\/wiki11\/archives\/19448","wordCount":23083,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4 \u8abf\u548c\u306e\u3068\u308c\u305f\u30b7\u30ea\u30fc\u30ba \u624b\u8db3\u3092\u5408\u8a08\u3059\u308b\u3053\u3068\u306b\u3088\u3063\u3066\u6570\u5b66\u306e\u30b7\u30ea\u30fc\u30ba\u3067\u3059 \u521d\u3081 \u3001 12\u3001 13\u3001 14\u3001 15\u3001 &#8230; {displaystyle 1\u3001{tfrac {1} {2}}\u3001{tfrac {1} {3}}\u3001{tfrac {1} {4}}\u3001{tfrac {1} {5}}\u3001dotc}}        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u8abf\u548c\u306e\u3068\u308c\u305f\u30b7\u30fc\u30b1\u30f3\u30b9\u304c\u767a\u751f\u3057\u307e\u3059\u3002\u3042\u306a\u305f\u306e\u90e8\u5206\u7684\u306a\u91d1\u984d\u3082\u305d\u3046\u3067\u3059 \u8abf\u548c\u306e\u3068\u308c\u305f\u6570 \u547c\u3073\u51fa\u3055\u308c\u307e\u3057\u305f\u3002\u3053\u308c\u3089\u306f\u3001\u305f\u3068\u3048\u3070\u3001\u7d44\u307f\u5408\u308f\u305b\u306e\u8cea\u554f\u3067\u4f7f\u7528\u3055\u308c\u3001\u30aa\u30a4\u30e9\u30fc\u30de\u30b7\u30a7\u30fc\u30cb\u5b9a\u6570\u3068\u5bc6\u63a5\u306b\u95a2\u9023\u3057\u3066\u3044\u307e\u3059 c {displaystyle\u30ac\u30f3\u30de} \u3002\u8abf\u548c\u306e\u3068\u308c\u305f\u30a8\u30d4\u30bd\u30fc\u30c9\u306f\u30bc\u30ed\u30b7\u30fc\u30b1\u30f3\u30b9\u3067\u3059\u304c\u3001\u8abf\u548c\u306e\u3068\u308c\u305f\u30b7\u30ea\u30fc\u30ba\u306f\u767a\u6563\u3057\u3066\u3044\u307e\u3059\u3002         (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4n {displaystyle n} -te partialsumme h n {displaystyle h_ {n}} \u8abf\u548c\u306e\u3068\u308c\u305f\u30b7\u30ea\u30fc\u30ba\u306f\u547c\u3070\u308c\u307e\u3059 n {displaystyle n} -e\u8abf\u548c\u306e\u6570\uff1a        (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4h n\uff1a= \u2211 k=1n1k= \u521d\u3081 + 12+ 13+ 14+ \u22ef + 1n{displaystyle h_ {n}\uff1a= sum _ {k = 1}^{n} {frac {1} {k}} = 1+ {frac {1} {2}}+{frac {1} {3}}+{frac {1} {1} {1} {1} {1} {1} {1}} {1} {1} {1} {1} {4}} {4}} {4}} {4}} \u8abf\u548c\u306e\u3068\u308c\u305f\u30b7\u30ea\u30fc\u30ba\u306f\u3001\u7279\u5225\u306a\u30b1\u30fc\u30b9\u3067\u3059 \u4e00\u822c\u7684\u306a\u8abf\u548c\u306e\u3068\u308c\u305f\u30b7\u30ea\u30fc\u30ba \u30b5\u30de\u30f3\u30c9\u3067 \u521d\u3081 \/ k a {displaystyle 1\/k^{alpha}} \u3001\u3053\u3053\u3067 a = \u521d\u3081 {displaystyle alpha = 1} \u3001 \u4e0b\u8a18\u53c2\u7167\u3002 \u3059\u3079\u3066\u306e\u30e1\u30f3\u30d0\u30fc\u306e\u305f\u3081\u306b\u3001\u8abf\u548c\u306e\u3068\u308c\u305f\u30b7\u30ea\u30fc\u30ba\u3068\u3044\u3046\u540d\u524d\u304c\u9078\u3070\u308c\u307e\u3057\u305f a k = 1k{displaystyle a_ {k} = {tfrac {1} {k}}} \u8abf\u548c\u306e\u3068\u308c\u305f\u624b\u6bb5 m h {displaystyle m_ {h}} 2\u4eba\u306e\u96a3\u63a5\u3059\u308b\u30e1\u30f3\u30d0\u30fc\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002 m H\uff08 a k\u22121\u3001 a k+1\uff09\uff09 = 21ak\u22121+1ak+1= 2k\u22121+k+1= 1k= a k{displaystyle m_ {text {h}}\uff08a_ {k-1}\u3001a_ {k+1}\uff09= {frac {2} {{frac {1} {a_ {k-1}}}}+{frac {1}} {a_ {k+1}}}}}}}}}} = a_ {k}}} Table of Contents\u6700\u521d\u306e\u90e8\u5206\u5408\u8a08\u306e\u5024 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u767a\u6563 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u6f38\u8fd1\u767a\u9054 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u7d71\u5408 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30c7\u30a3\u30ac\u30f3\u30de\u95a2\u6570\u3068\u306e\u95a2\u4fc2 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u5c0e\u51fa [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30bc\u30ed\u304b\u30891\u3078\u306e\u7d71\u5408 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] Exponential\u306e\u830e\u6a5f\u80fd\u306e\u524d\u63d0 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u751f\u6210\u6a5f\u80fd [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u8abf\u548c\u306e\u3068\u308c\u305f\u6570\u306b\u3064\u3044\u3066\u306e\u884c [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30d1\u30ef\u30fc\u30d9\u30af\u30c8\u30eb [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30de\u30c3\u30b7\u30e5\u5217 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u4ea4\u4e92\u306e\u9ad8\u8abf\u6ce2\u30b7\u30ea\u30fc\u30ba [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u4e00\u822c\u7684\u306a\u9ad8\u8abf\u6ce2\u30b7\u30ea\u30fc\u30ba [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u30b5\u30d6\u30cf\u30fc\u30e2\u30cb\u30c3\u30af\u306a\u5217 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u6700\u521d\u306e\u90e8\u5206\u5408\u8a08\u306e\u5024 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] H1=1H2=32=1,5H3=116=1,83\u00afH4=2512=2,083\u00afH5=13760=2,283\u00afH6=4920=2,45H7=363140=2,59285714\u00afH8=761280=2,717857142\u00afH9=71292520=2,828968253\u00afH10=73812520=2,928968253\u00af{displayStyle {begin {matrix} h_ {1}\uff06=\uff061 \\\\ h_ {2}\uff06=\uff06{frac {3} {2}}}}}}\uff061 {\u3001} 4}\uff06=\uff06{frac {25} {12}}\uff06=\uff062 {\u3001} 08 {\u30aa\u30fc\u30d0\u30fc\u30e9\u30a4\u30f3{3}} \\\\ h_ {5}\uff06=\uff06{frac {137} {60}}}\uff06=\uff06=\uff062 {\u3001} 28 {overline } h_ {6}\uff06=\uff06{frac {49} {20} {20}}\uff06=\uff062 {\u3001} 45 \\\\ h_ {7}\uff06=\uff06{frac {363} {140}}\uff06=\uff062 {\u3001} 59 {anuverline 0}}\uff06=\uff062 {\u3001} 717 {overline {857142}} \\\\ h_ {9}\uff06=\uff06{frac {7129} {2520}}\uff06=\uff062 {\u3001} 828 {anuverline {968253}}} \\ \\ h_ {{frac}} \\ \\ \\ \\ \\ H_ }}\uff06=\uff062 {\u3001} 928 {overline {968253}} \\\\ end {matrix}}}} \u306e\u5206\u6bcd h n {displaystyle h_ {n}} \u3059\u3079\u3066\u306e\u7d20\u6570\u306e\u52b9\u529b\u3092\u4ecb\u3057\u3066\u3044\u307e\u3059 p k {displaystyle p^{k}} \u3068 n \/ 2 < p k \u2264 n {displaystyle n\/2 h n {displaystyle h_ {m} -h_ {n}} \u305f\u3081\u306b m \u2260 n {displaystyle mneq n} \u6574\u6570\u306f\uff08K\u00fcrsch\u00e1k1918\uff09\u3067\u3059\u3002 [2] \u3053\u308c\u306f\u3001Nagell 1923\u304b\u3089\u306e\u6587\u306e\u7279\u5225\u306a\u30b1\u30fc\u30b9\u3067\u3059\u3002 [3] \u306f p \u2265 5 {displaystyle pgeq 5} \u30d7\u30e9\u30a4\u30e0\u30ca\u30f3\u30d0\u30fc\u3001\u305d\u308c\u304c\u30ab\u30a6\u30f3\u30bf\u30fc\u3067\u3059 h p  &#8211;  \u521d\u3081 {displaystyle h_ {p-1}} Woltstenholme\u306e\u5224\u6c7a\u306b\u3088\u308b\u3068 p 2 {displaystyle p^{2}} \u5206\u88c2\u53ef\u80fd\u3067\u3059 p {displaystyle p} Woltstenholme Prim\u3001\u305d\u308c\u304b\u3089\u3055\u3048 p 3 {displaystyle p^{3}} \u3002  \u767a\u6563 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u8abf\u548c\u306e\u3068\u308c\u305f\u30b7\u30ea\u30fc\u30ba\u306f\u3001\u6700\u521d\u306b\u30aa\u30ec\u30ba\u30e1\uff0814\u4e16\u7d00\uff09\u306e\u30cb\u30b3\u30e9\u30b9\u304c\u5b9f\u8a3c\u3057\u305f\u3088\u3046\u306b\u3001\u7121\u9650\u3068\u306f\u7570\u306a\u308a\u307e\u3059\u3002\u3053\u308c\u306f\u3001\u3059\u3079\u3066\u306e\u30e1\u30f3\u30d0\u30fc\u3067\u5c0f\u3055\u3044\u307e\u305f\u306f\u540c\u3058\u884c\u3068\u6bd4\u8f03\u3059\u308b\u3053\u3068\u3067\u898b\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\uff08\u672a\u6210\u5e74\u8005\u306e\u57fa\u6e96\uff09\uff1a Hn=\u00a01\u00a0+\u00a01\/2+(1\/3+1\/4)+(1\/5+1\/6+1\/7+1\/8)+\u00a0\u22ef\u00a0+\u00a01\/n\u2265\u00a01\u00a0+\u00a01\/2+(1\/4+1\/4)+(1\/8+1\/8+1\/8+1\/8)+\u00a0\u22ef\u00a0+\u00a01\/n=\u00a01\u00a0+\u00a01\/2+1\/2+1\/2+\u00a0\u22ef\u00a0+\u00a01\/n{displaystyle {begin {matrix} h_ {n}\uff06= 1+1\/2\uff06+\uff06\uff081\/3+1\/4\uff09\uff06+\uff06\uff081\/5+1\/6+1\/7+1\/8\uff09\uff06+1\/n \\\uff06geq 1+1\/2\uff06+\uff06\uff081\/4+1\/4\uff09 = 1+ 1\/2\uff06+\uff061\/2\uff06+\uff061\/2\uff06+++ cdots+ 1\/nend {matrix}}}} \u6700\u5f8c\u306e\u884c\u306e\u5408\u8a08\u306f\u3001\u3059\u3079\u3066\u306e\u5024\u3092\u8d85\u3048\u3066\u3044\u307e\u3059 n {displaystyle n} \u5341\u5206\u3067\u3059\u3002\u3088\u308a\u6b63\u78ba\u306b\u898b\u7a4d\u3082\u308a\u3092\u53d6\u5f97\u3057\u307e\u3059 h 2\u2113\u2265 \u521d\u3081 + \u2113 \/ 2 {displaystyle h_ {2^{ell}} geq 1+ell \/2} \u305f\u3081\u306b \u2113 = 0 \u3001 \u521d\u3081 \u3001 2 \u3001 &#8230; {displaystyle ell = 0.1.2\u3001dots} \u6f38\u8fd1\u767a\u9054 [ \u7de8\u96c6 | \u30bd\u30fc\u30b9\u30c6\u30ad\u30b9\u30c8\u3092\u7de8\u96c6\u3057\u307e\u3059 ] \u6f38\u8fd1\u958b\u767a\u304c\u9069\u7528\u3055\u308c\u307e\u3059\uff1a Hn=\u2211k=1n1k=ln\u2061n+\u03b3+12n\u2212112n2+1120n4\u22121252n6+1240n8\u22121132n10+O(1n12)=ln\u2061n+\u03b3+O(1n),n\u2192\u221e{displaystyle {begin{aligned}H_{n}=sum _{k=1}^{n}{frac {1}{k}}&#038;=ln n+gamma +{frac {1}{2n}}-{frac {1}{12n^{2}}}+{frac {1}{120n^{4}}}-{frac {1}{252n^{6}}}+{frac {1}{240n^{8}}}-{frac {1}{132n^{10}}}+{mathcal {O}}!left({frac {1}{n^{12}}}right)\\&#038;=ln n+gamma +{mathcal {O}}!left({frac {1}{n}}right),quad nto infty end{aligned}}} \u3053\u3061\u3089\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044 ln \u2061 n {displaystyle ln n} \u81ea\u7136\u5bfe\u6570\u3068\u30e9\u30f3\u30c0\u30a6\u306e\u30b7\u30f3\u30dc\u30eb o {displaystyle {mathcal {o}}} \u958b\u767a\u306e\u305f\u3081\u306e\u958b\u767a\u306e\u6b8b\u308a\u306e\u884c\u52d5\u306e\u884c\u52d5\u306b\u3064\u3044\u3066\u8aac\u660e\u3057\u307e\u3059 n \u2192 \u221e {displaystyle nto infty} \u3002\u6570\u5b66\u5b9a\u6570 c {displaystyle\u30ac\u30f3\u30de}  \uff08\u30ac\u30f3\u30de\uff09 Euler-Mascheroni\u5b9a\u6570\u3068\u305d\u306e\u6570\u5024\u306f0.5772156649\u3067\u3059&#8230;  \u8fd1\u4f3cLn\u3092\u5099\u3048\u305f\u8abf\u548c\u7cfb\u30b7\u30ea\u30fc\u30ba\u306e\u90e8\u5206\u7684\u5408\u8a08 n + c \u304a\u3088\u3073LN\u3092\u63a8\u5b9a\u3057\u307e\u3059 n +1 \u3055\u3089\u306b\u3001\u4ee5\u4e0b\u304c\u9069\u7528\u3055\u308c\u307e\u3059 h n < ln \u2061 n + \u521d\u3081 {displaystyle h_ {n} 011\u2212yn1\u2212yd \u3068 = \u222b 01\uff08 \u521d\u3081 + \u3068 + \u22ef + \u3068 n\u22121\uff09\uff09  d \u3068 = \u521d\u3081 + 12+ \u22ef + 1n= h n{displaystyle int _ {0}^{1} {frac {1-y^{n}} {1-y}}\u3001mathrm {d} y = int _ {0}^{1}\uff081+y+cdots+y^{n-1}\uff09Mathrm {d} y = 1+{1} {1} {1} {1} {n}} = h_ {n}} \u3053\u306e\u8868\u73fe\u306f\u4e00\u822c\u5316\u3057\u307e\u3059 n {displaystyle n} -te\u306e\u8907\u96d1\u306a\u5024\u306b\u95a2\u3059\u308b\u8abf\u548c\u306e\u3068\u308c\u305f\u6570 n {displaystyle n} \u3068 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