. . "Een Kempnerreeks of reeks van Kempner ontstaat door in de harmonische reeks alle termen waarvan de noemer een bepaald cijfer bevat, te schrappen. Door bijvoorbeeld alle termen met in de noemer het cijfer 9, te schrappen, ontstaat de Kempnerreeks: Van de eerste honderd termen van de harmonische reeks werden dus verwijderd: tot en met . Deze reeks werd voor het eerst bestudeerd door A. J. Kempner in 1914. De reeks is merkwaardig omdat ze, tegen de intu\u00EFtie ingaand, convergent is. Kempner bewees dat de som kleiner is dan 80, en Baillie toonde aan de som, wat de eerste 20 decimalen betreft, gelijk is aan 22,92067 66192 64150 34816. Schmelzer en Baillie vonden een effici\u00EBnt algoritme voor het meer algemene probleem, waarbij alle termen uit de harmonische reeks worden verwijderd die een bepaalde eindige string van cijfers bevatten. Bijvoorbeeld, indien alle noemers waarin de string \"42\" voorkomt worden verwijderd is de som afgerond 228,44630 41592 30813 25415. Indien de string \"314159\" wordt verwijderd is de som afgerond 2302582,33386 37826 07892 02376."@nl . "In der Mathematik bezeichnen die zehn Kempner-Reihen, benannt nach Aubrey J. Kempner, diejenigen Reihen, die dadurch entstehen, dass man aus der harmonischen Reihe alle Summanden entfernt, die eine bestimmte dezimale Ziffer in ihrem Nenner enthalten. Die Kempner-Reihen geh\u00F6ren daher zu den subharmonischen Reihen. L\u00E4sst man etwa alle Summanden weg, deren Nenner die Ziffer in seiner Dezimalschreibweise enth\u00E4lt, ergibt sich die Kempner-Reihe als Oder durch Auslassen der Summanden mit einer im Nenner: Sie wurden erstmals von Aubrey J. Kempner 1914 beschrieben."@de . . . "8982"^^ . "Kempnerreeks"@nl . . . . "Kempner series"@en . . "S\u00E9rie de Kempner"@fr . "\u80AF\u666E\u7D0D\u7D1A\u6578\uFF08\u82F1\u8A9E\uFF1AKempner series\uFF09\u662F\u5341\u9032\u5236\u5BEB\u6CD5\u4E0D\u542B\u6578\u5B579\u7684\u6B63\u6574\u6578\u7684\u5012\u6578\u548C\u3002\u7528\u7B26\u865F\u53EF\u5BEB\u6210 \u5176\u4E2D\u300C\u7F3A9\u300D\u610F\u601D\u662F\u300C\u5341\u9032\u5236\u8868\u793A\u4E2D\uFF0C\u4E0D\u542B\u6578\u5B579\u300D\uFF0C\u4E0B\u540C\u3002\u65BC1914\u5E74\u6700\u65E9\u7814\u7A76\u8A72\u7D1A\u6578\u3002\u80AF\u666E\u7D0D\u7D1A\u6578\u662F\u7531\u8ABF\u548C\u7D1A\u6578\u522A\u8D70\u542B\u6578\u5B579\u7684\u9805\u6240\u5F97\uFF0C\u4F46\u80AF\u666E\u7D0D\u7D1A\u6578\u6536\u6582\uFF0C\u8ABF\u548C\u7D1A\u6578\u5247\u767C\u6563\u3002\u80AF\u666E\u7D0D\u8B49\u660E\uFF0C\u7D1A\u6578\u4E4B\u548C\u5C0F\u65BC90\u3002\u7F85\u4F2F\u7279\u00B7\u8C9D\u5229\u8B49\u660E\uFF0C\u7D1A\u6578\u6E96\u78BA\u5230\u5C0F\u6578\u9EDE\u5F8C20\u4F4D\u7684\u503C\u70BA22.92067661926415034816\uFF08OEIS\u6578\u5217\uFF09\u3002 \u76F4\u89C0\u7406\u89E3\uFF0C\u7D1A\u6578\u6536\u6582\u662F\u56E0\u70BA\u5927\u90E8\u5206\u300C\u5927\u6578\u300D\u90FD\u6709\u9F4A0\u81F39\u7684\u5168\u90E8\u6578\u5B57\u3002\u4F8B\u5982\uFF0C\u5747\u52FB\u96A8\u6A5F\u9078\u4E00\u500B100\u4F4D\u7684\u6B63\u6574\u6578\uFF0C\u5F88\u6613\u5305\u542B\u81F3\u5C11\u4E00\u500B\u6578\u5B579\uFF0C\u65BC\u662F\u7D1A\u6578\u4E0D\u8A08\u8A72\u6578\u7684\u5012\u6578\u3002 \u65BD\u6885\u723E\u7B56\u8207\u8C9D\u5229\u627E\u5230\u9AD8\u6548\u7B97\u6CD5\uFF0C\u7D66\u5B9A\u4EFB\u610F\u6578\u5B57\u4E32\u70BA\u8F38\u5165\uFF0C\u8A08\u7B97\u7F3A\u8A72\u4E32\u7684\u6B63\u6574\u6578\u5012\u6578\u548C\u3002\u6B64\u554F\u984C\u63A8\u5EE3\u4E86\u539F\u672C\u7684\u7D1A\u6578\u6C42\u503C\u554F\u984C\u3002\u8209\u4F8B\uFF0C\u8003\u616E\u6240\u6709\u7F3A\u6578\u5B57\u4E32\u300C42\u300D\u7684\u6B63\u6574\u6578\uFF0C\u5176\u5012\u6578\u548C\u7D04\u70BA228.44630415923081325415\u3002\u53C8\u8209\u4F8B\uFF0C\u7F3A\u6578\u5B57\u4E32\u300C314159\u300D\u7684\u6B63\u6574\u6578\u5012\u6578\u548C\u7D04\u70BA2302582.33386378260789202376\u3002\uFF08\u4E0A\u8FF0\u6578\u503C\u7686\u56DB\u6368\u4E94\u5165\u81F3\u672B\u4F4D\u3002\uFF09"@zh . . . . . "The Kempner series is a modification of the harmonic series, formed by omitting all terms whose denominator expressed in base 10 contains the digit 9. That is, it is the sum where the prime indicates that n takes only values whose decimal expansion has no nines. The series was first studied by A. J. Kempner in 1914. The series is counterintuitive because, unlike the harmonic series, it converges. Kempner showed the sum of this series is less than 90. Baillie showed that, rounded to 20 decimals, the actual sum is 22.92067661926415034816(sequence in the OEIS)."@en . . . . "1122974105"^^ . . . . . "Serie di Kempner"@it . "La serie di Kempner \u00E8 una variante della serie armonica, costruita omettendo tutti i termini il cui denominatore contiene la cifra espressa in base decimale. Cio\u00E8, \u00E8 la somma dove l'apice indica che assume solo i valori la cui espansione decimale non contiene dei . La serie fu per la prima volta studiata da A. J. Kempner nel 1914. La serie \u00E8 interessante a causa del risultato controintuitivo che, a differenza della serie armonica, la serie di Kempner converge. Kempner mostr\u00F2 che il valore di questa serie \u00E8 minore di . Baillie dimostr\u00F2 che, arrotondata alla 20\u00AA cifra decimale, la somma reale \u00E8 . Euristicamente, questa serie converge perch\u00E9 gli interi molto grandi hanno pi\u00F9 probabilit\u00E0 di possedere qualunque cifra. Per esempio, \u00E8 davvero molto probabile che un intero casuale di cifre contenga almeno un , causandone l'esclusione dalla precedente somma. Schmelzer e Baillie trovarono un algoritmo efficiente per il problema dell'omissione di stringhe di cifre. Per esempio, la somma di dove non contiene \"42\" \u00E8 all'incirca . Un altro esempio: la somma di dove in non appare la stringa \"314159\" (le prime cifre del \u03C0) \u00E8 approssimativamente ."@it . . . . "17961670"^^ . . "La s\u00E9rie de Kempner est une s\u00E9rie obtenue \u00E0 partir de la s\u00E9rie harmonique en excluant tous les termes dont le d\u00E9nominateur, exprim\u00E9 en base dix, contient le chiffre 9. La somme des termes de cette s\u00E9rie s'\u00E9crit : o\u00F9 le prime dans signifie que n ne prend que les valeurs dont le d\u00E9veloppement d\u00E9cimal ne contient pas de 9."@fr . "La serie de Kempner es una modificaci\u00F3n de la serie arm\u00F3nica, en la cual se omiten todos los t\u00E9rminos cuyo denominador expresado en base 10 contiene al menos un d\u00EDgito 9, es decir, es la serie donde la prima indica que toma solo valores cuya expresi\u00F3n en base decimal no contiene ning\u00FAn 9. Esta serie fue estudiada por en 1914.\u200B Esta serie es interesante porque, al contrario que la serie arm\u00F3nica y contra-intuitivamente, es una serie convergente (Kempner demostr\u00F3 que su valor es menor que 80, y Baillie\u200B showed obtuvo su resultado con una precisi\u00F3n de 20 decimales. El resultado de la serie es 22.92067 66192 64150 34816(sucesi\u00F3n A082838 en OEIS)). Schmelzer y Baillie\u200B obtuvieron un algoritmo eficiente para el problema m\u00E1s general de resolver series en las que se omitieran sumandos que contuvierancualquier cadena dada de d\u00EDgitos. Por ejemplo, la suma de para los que no contengan la cadena \"42\" en su expresi\u00F3n decimal es 228.44630 41592 30813 25415. Otro ejemplo m\u00E1s complicado, en el que se calcula la suma de para los que no contengan la cadena \"314159\" es 2302582.33386 37826 07892 02376. (Todos los valores num\u00E9ricos aqu\u00E7\u00ED dados est\u00E1n redondeados en su \u00FAltima cifra decimal)."@es . "La serie de Kempner es una modificaci\u00F3n de la serie arm\u00F3nica, en la cual se omiten todos los t\u00E9rminos cuyo denominador expresado en base 10 contiene al menos un d\u00EDgito 9, es decir, es la serie donde la prima indica que toma solo valores cuya expresi\u00F3n en base decimal no contiene ning\u00FAn 9. Esta serie fue estudiada por en 1914.\u200B Esta serie es interesante porque, al contrario que la serie arm\u00F3nica y contra-intuitivamente, es una serie convergente (Kempner demostr\u00F3 que su valor es menor que 80, y Baillie\u200B showed obtuvo su resultado con una precisi\u00F3n de 20 decimales. El resultado de la serie es 22.92067 66192 64150 34816(sucesi\u00F3n A082838 en OEIS))."@es . . . . "\u80AF\u666E\u7D0D\u7D1A\u6578\uFF08\u82F1\u8A9E\uFF1AKempner series\uFF09\u662F\u5341\u9032\u5236\u5BEB\u6CD5\u4E0D\u542B\u6578\u5B579\u7684\u6B63\u6574\u6578\u7684\u5012\u6578\u548C\u3002\u7528\u7B26\u865F\u53EF\u5BEB\u6210 \u5176\u4E2D\u300C\u7F3A9\u300D\u610F\u601D\u662F\u300C\u5341\u9032\u5236\u8868\u793A\u4E2D\uFF0C\u4E0D\u542B\u6578\u5B579\u300D\uFF0C\u4E0B\u540C\u3002\u65BC1914\u5E74\u6700\u65E9\u7814\u7A76\u8A72\u7D1A\u6578\u3002\u80AF\u666E\u7D0D\u7D1A\u6578\u662F\u7531\u8ABF\u548C\u7D1A\u6578\u522A\u8D70\u542B\u6578\u5B579\u7684\u9805\u6240\u5F97\uFF0C\u4F46\u80AF\u666E\u7D0D\u7D1A\u6578\u6536\u6582\uFF0C\u8ABF\u548C\u7D1A\u6578\u5247\u767C\u6563\u3002\u80AF\u666E\u7D0D\u8B49\u660E\uFF0C\u7D1A\u6578\u4E4B\u548C\u5C0F\u65BC90\u3002\u7F85\u4F2F\u7279\u00B7\u8C9D\u5229\u8B49\u660E\uFF0C\u7D1A\u6578\u6E96\u78BA\u5230\u5C0F\u6578\u9EDE\u5F8C20\u4F4D\u7684\u503C\u70BA22.92067661926415034816\uFF08OEIS\u6578\u5217\uFF09\u3002 \u76F4\u89C0\u7406\u89E3\uFF0C\u7D1A\u6578\u6536\u6582\u662F\u56E0\u70BA\u5927\u90E8\u5206\u300C\u5927\u6578\u300D\u90FD\u6709\u9F4A0\u81F39\u7684\u5168\u90E8\u6578\u5B57\u3002\u4F8B\u5982\uFF0C\u5747\u52FB\u96A8\u6A5F\u9078\u4E00\u500B100\u4F4D\u7684\u6B63\u6574\u6578\uFF0C\u5F88\u6613\u5305\u542B\u81F3\u5C11\u4E00\u500B\u6578\u5B579\uFF0C\u65BC\u662F\u7D1A\u6578\u4E0D\u8A08\u8A72\u6578\u7684\u5012\u6578\u3002 \u65BD\u6885\u723E\u7B56\u8207\u8C9D\u5229\u627E\u5230\u9AD8\u6548\u7B97\u6CD5\uFF0C\u7D66\u5B9A\u4EFB\u610F\u6578\u5B57\u4E32\u70BA\u8F38\u5165\uFF0C\u8A08\u7B97\u7F3A\u8A72\u4E32\u7684\u6B63\u6574\u6578\u5012\u6578\u548C\u3002\u6B64\u554F\u984C\u63A8\u5EE3\u4E86\u539F\u672C\u7684\u7D1A\u6578\u6C42\u503C\u554F\u984C\u3002\u8209\u4F8B\uFF0C\u8003\u616E\u6240\u6709\u7F3A\u6578\u5B57\u4E32\u300C42\u300D\u7684\u6B63\u6574\u6578\uFF0C\u5176\u5012\u6578\u548C\u7D04\u70BA228.44630415923081325415\u3002\u53C8\u8209\u4F8B\uFF0C\u7F3A\u6578\u5B57\u4E32\u300C314159\u300D\u7684\u6B63\u6574\u6578\u5012\u6578\u548C\u7D04\u70BA2302582.33386378260789202376\u3002\uFF08\u4E0A\u8FF0\u6578\u503C\u7686\u56DB\u6368\u4E94\u5165\u81F3\u672B\u4F4D\u3002\uFF09"@zh . "Serie de Kempner"@es . . . . "In der Mathematik bezeichnen die zehn Kempner-Reihen, benannt nach Aubrey J. Kempner, diejenigen Reihen, die dadurch entstehen, dass man aus der harmonischen Reihe alle Summanden entfernt, die eine bestimmte dezimale Ziffer in ihrem Nenner enthalten. Die Kempner-Reihen geh\u00F6ren daher zu den subharmonischen Reihen. L\u00E4sst man etwa alle Summanden weg, deren Nenner die Ziffer in seiner Dezimalschreibweise enth\u00E4lt, ergibt sich die Kempner-Reihe als Oder durch Auslassen der Summanden mit einer im Nenner: Sie wurden erstmals von Aubrey J. Kempner 1914 beschrieben. Das Interessante an diesen zehn Reihen ist, dass sie alle konvergieren, obwohl die harmonische Reihe selbst nicht konvergiert. Dies wurde von Kempner bewiesen; daher werden die Reihen oft Kempner-Reihen genannt.Die Konvergenzeigenschaft wird auch dadurch deutlich, dass bereits ab 7-stelligen Zahlen diese mehrheitlich wegfallen und es bei gro\u00DFen Zahlen nur wenige gibt, die eine bestimmte Ziffer nicht enthalten und so einen Additionsbeitrag leisten k\u00F6nnen."@de . "La s\u00E9rie de Kempner est une s\u00E9rie obtenue \u00E0 partir de la s\u00E9rie harmonique en excluant tous les termes dont le d\u00E9nominateur, exprim\u00E9 en base dix, contient le chiffre 9. La somme des termes de cette s\u00E9rie s'\u00E9crit : o\u00F9 le prime dans signifie que n ne prend que les valeurs dont le d\u00E9veloppement d\u00E9cimal ne contient pas de 9. Son int\u00E9r\u00EAt r\u00E9side dans le fait que contrairement \u00E0 la s\u00E9rie harmonique, elle converge. Ce r\u00E9sultat fut d\u00E9montr\u00E9 en 1914 par (en). Mais il fallut attendre la fin des ann\u00E9es 1970 pour qu'on en d\u00E9termine une valeur approch\u00E9e de la somme au moyen de m\u00E9thodes astucieuses, en raison de sa tr\u00E8s lente vitesse de convergence."@fr . "The Kempner series is a modification of the harmonic series, formed by omitting all terms whose denominator expressed in base 10 contains the digit 9. That is, it is the sum where the prime indicates that n takes only values whose decimal expansion has no nines. The series was first studied by A. J. Kempner in 1914. The series is counterintuitive because, unlike the harmonic series, it converges. Kempner showed the sum of this series is less than 90. Baillie showed that, rounded to 20 decimals, the actual sum is 22.92067661926415034816(sequence in the OEIS). Heuristically, this series converges because most large integers contain every digit. For example, a random 100-digit integer is very likely to contain at least one '9', causing it to be excluded from the above sum. Schmelzer and Baillie found an efficient algorithm for the more general problem of any omitted string of digits. For example, the sum of 1/n where n has no instances of \"42\" is about 228.44630415923081325415. Another example: the sum of 1/n where n has no occurrence of the digit string \"314159\" is about 2302582.33386378260789202376. (All values are rounded in the last decimal place.)"@en . "\u80AF\u666E\u7D0D\u7D1A\u6578"@zh . . "Een Kempnerreeks of reeks van Kempner ontstaat door in de harmonische reeks alle termen waarvan de noemer een bepaald cijfer bevat, te schrappen. Door bijvoorbeeld alle termen met in de noemer het cijfer 9, te schrappen, ontstaat de Kempnerreeks: Van de eerste honderd termen van de harmonische reeks werden dus verwijderd: tot en met . Deze reeks werd voor het eerst bestudeerd door A. J. Kempner in 1914."@nl . . "Kempner-Reihe"@de . "La serie di Kempner \u00E8 una variante della serie armonica, costruita omettendo tutti i termini il cui denominatore contiene la cifra espressa in base decimale. Cio\u00E8, \u00E8 la somma dove l'apice indica che assume solo i valori la cui espansione decimale non contiene dei . La serie fu per la prima volta studiata da A. J. Kempner nel 1914. La serie \u00E8 interessante a causa del risultato controintuitivo che, a differenza della serie armonica, la serie di Kempner converge. Kempner mostr\u00F2 che il valore di questa serie \u00E8 minore di . Baillie dimostr\u00F2 che, arrotondata alla 20\u00AA cifra decimale, la somma reale \u00E8 ."@it . . .