. . . . . . . . . . . . . . "En matematiko, serio estas sumo da eroj el vico de nombroj. Por donita vico , la n-a Sn estas sumo de la unuaj n eroj de la vico, tio estas, Serio estas konver\u011Da se kaj nur se la vico de \u011Diaj partaj sumoj konver\u011Das (ekzistas limigo de vico). En pli formala lingvo, serio konver\u011Das se tie ekzistas limigo y, tia ke por \u0109iu ajne malgranda pozitiva nombro e, e>0, estas entjero N tia ke por \u0109iuj n \u2265 N, Serio kiu ne estas konver\u011Da estas malkonver\u011Da serio (anka\u016D nomita diver\u011Da serio)."@eo . . "In matematica, una serie convergente \u00E8 una serie tale che il limite delle sue somme parziali \u00E8 finito. Questo vuol dire che, data una successione , la serie \u00E8 convergente se la successione delle somme parziali ha un limite finito, cio\u00E8 se esiste finito tale che per ogni esiste tale che per ogni Il numero \u00E8 detto somma della serie: spesso \u00E8 difficile trovare questo numero, sebbene possa essere facile capire che una serie \u00E8 convergente. La somma di due serie convergenti \u00E8 ovviamente ancora convergente, cos\u00EC come la serie prodotta dalla moltiplicazione di una serie per uno scalare; le serie convergenti formano quindi uno spazio vettoriale sul campo dei numeri reali. Una serie non convergente non \u00E8 necessariamente detta divergente, ad esempio la serie non \u00E8 n\u00E9 convergente n\u00E9 divergente, in quanto la sua successione delle somme parziali oscilla tra i valori e e quindi non ammette limite."@it . . . . . "\u0645\u062A\u0633\u0644\u0633\u0644\u0629 \u0645\u062A\u0642\u0627\u0631\u0628\u0629"@ar . . "In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence defines a series S that is denoted The nth partial sum Sn is the sum of the first n terms of the sequence; that is, If the series is convergent, the (necessarily unique) number is called the sum of the series. The same notation is used for the series, and, if it is convergent, to its sum. This convention is similar to that which is used for addition: a + b denotes the operation of adding a and b as well as the result of this addition, which is called the sum of a and b."@en . "S\u00E9rie convergente"@pt . . . . . "\u6570\u5B66\u306B\u304A\u3044\u3066\u3001\u53CE\u675F\u7D1A\u6570 (convergent series) \u3068\u306F\u3001\u305D\u306E\u90E8\u5206\u548C\u306E\u6210\u3059\u6570\u5217\u304C\u53CE\u675F\u3059\u308B\u3088\u3046\u306A\u7D1A\u6570\u3067\u3042\u308B\u3002 \u3053\u3053\u3067\u3001\u7D1A\u6570\u3068\u306F\u6570\u5217\u306E\u9805\u306E\u7DCF\u548C\u306E\u3053\u3068\u3067\u3042\u308A\u3001\u4E0E\u3048\u3089\u308C\u305F\u6570\u5217 a1, a2, ..., an, ... \u306E\u7B2C n-\u90E8\u5206\u548C\u3068\u306F\u6700\u521D\u306E n-\u9805\u306E\u6709\u9650\u548C \u306E\u3053\u3068\u3092\u6307\u3059\u3002 \u3042\u308B\u7D1A\u6570\u304C\u53CE\u675F\u7D1A\u6570\u3067\u3042\u308B\u3053\u3068\u306F\u3001\u300C\uFF08\u6709\u9650\u306A\uFF09\u548C\u3092\u6301\u3064\u300D\u3068\u304B\u300C\u548C\u304C\u6709\u9650\u78BA\u5B9A\u3067\u3042\u308B\u300D\u306A\u3069\u3068\u8A00\u3044\u8868\u3055\u308C\u308B\u3002"@ja . . . "Serie konbergente"@eu . . . "Konver\u011Da serio"@eo . . . "Convergentie (wiskunde)"@nl . . . "p/s084670"@en . . "1743431"^^ . . "En matematiko, serio estas sumo da eroj el vico de nombroj. Por donita vico , la n-a Sn estas sumo de la unuaj n eroj de la vico, tio estas, Serio estas konver\u011Da se kaj nur se la vico de \u011Diaj partaj sumoj konver\u011Das (ekzistas limigo de vico). En pli formala lingvo, serio konver\u011Das se tie ekzistas limigo y, tia ke por \u0109iu ajne malgranda pozitiva nombro e, e>0, estas entjero N tia ke por \u0109iuj n \u2265 N, Serio kiu ne estas konver\u011Da estas malkonver\u011Da serio (anka\u016D nomita diver\u011Da serio)."@eo . "Series"@en . . . . "In de wiskunde is convergentie een eigenschap van sommige rijen dat naarmate men verder in de rij komt de elementen van de rij een bepaalde waarde blijken te naderen. Zo'n rij heet convergent en de benaderde waarde wordt de limiet van de rij genoemd. De termen (getallen) in de rij heten convergenten. Zo convergeert de rij overduidelijk naar de limiet 0. In de wiskunde is men nog iets preciezer, en wordt de genoemde rij bijvoorbeeld niet als convergent beschouwd in de positieve getallen, omdat er geen positief getal is waarnaar de rij streeft."@nl . . "Em matem\u00E1tica, uma s\u00E9rie \u00E9 o somat\u00F3rio dos termos de uma sequ\u00EAncia de n\u00FAmeros. Dada uma sequ\u00EAncia infinita , a -\u00E9sima soma parcial \u00E9 a soma dos primeiros termos da sequ\u00EAncia, isto \u00E9, Uma s\u00E9rie \u00E9 convergente se a sequ\u00EAncia de suas somas parciais tende a um limite. Isto quer dizer que as somas parciais se tornam cada vez mais pr\u00F3ximas de um dado n\u00FAmero quando o n\u00FAmero de seus termos aumenta. Em uma linguagem mais formal, uma s\u00E9rie converge se existe um limite tal que para qualquer n\u00FAmero positivo arbitrariamente pequeno , existe um inteiro tal que para todo , Qualquer s\u00E9rie que n\u00E3o \u00E9 convergente \u00E9 chamada de divergente."@pt . . . . . "S\u00E9rie convergente"@fr . "En matem\u00E0tiques, una s\u00E8rie \u00E9s la suma dels termes d'una successi\u00F3 infinita de nombres. Donada una seq\u00FC\u00E8ncia infinita , l'en\u00E8ssima suma parcial \u00E9s la suma dels primers n termes de la seq\u00FC\u00E8ncia, \u00E9s a dir: Una s\u00E8rie \u00E9s convergent si la seq\u00FC\u00E8ncia de les seves sumes parcials tendeix a un l\u00EDmit; \u00E9s a dir que les sumes parcials s'acosten m\u00E9s i m\u00E9s a un determinat nombre quan el nombre de termes augmenta. M\u00E9s precisament, una s\u00E8rie convergeix si existeix un nombre tal que per qualsevol nombre positiu petit i arbitrari , existeix un enter suficientment gran tal que per tot ,"@ca . . . . "En matem\u00E1ticas, una serie (suma de los t\u00E9rminos de una secuencia de n\u00FAmeros), resulta convergente si la sucesi\u00F3n de sumas parciales tiene un l\u00EDmite en el espacio considerado. De otro modo, constituir\u00EDa lo que se denomina serie divergente."@es . . "\u53CE\u675F\u7D1A\u6570"@ja . "En math\u00E9matiques, une s\u00E9rie est dite convergente si la suite de ses sommes partielles a une limite dans l'espace consid\u00E9r\u00E9. Dans le cas contraire, elle est dite divergente. Pour des s\u00E9ries num\u00E9riques, ou \u00E0 valeurs dans un espace de Banach \u2014 c'est-\u00E0-dire un espace vectoriel norm\u00E9 complet \u2014, il suffit de prouver la convergence absolue de la s\u00E9rie pour montrer sa convergence, ce qui permet de se ramener \u00E0 une s\u00E9rie \u00E0 termes r\u00E9els positifs. Pour \u00E9tudier ces derni\u00E8res, il existe une large vari\u00E9t\u00E9 de r\u00E9sultats, tous fond\u00E9s sur le principe de comparaison."@fr . . . . . . . . . . . . . . . . . "Em matem\u00E1tica, uma s\u00E9rie \u00E9 o somat\u00F3rio dos termos de uma sequ\u00EAncia de n\u00FAmeros. Dada uma sequ\u00EAncia infinita , a -\u00E9sima soma parcial \u00E9 a soma dos primeiros termos da sequ\u00EAncia, isto \u00E9, Uma s\u00E9rie \u00E9 convergente se a sequ\u00EAncia de suas somas parciais tende a um limite. Isto quer dizer que as somas parciais se tornam cada vez mais pr\u00F3ximas de um dado n\u00FAmero quando o n\u00FAmero de seus termos aumenta. Em uma linguagem mais formal, uma s\u00E9rie converge se existe um limite tal que para qualquer n\u00FAmero positivo arbitrariamente pequeno , existe um inteiro tal que para todo ,"@pt . . . . . . . . . . "1071654886"^^ . . . "En matem\u00E0tiques, una s\u00E8rie \u00E9s la suma dels termes d'una successi\u00F3 infinita de nombres. Donada una seq\u00FC\u00E8ncia infinita , l'en\u00E8ssima suma parcial \u00E9s la suma dels primers n termes de la seq\u00FC\u00E8ncia, \u00E9s a dir: Una s\u00E8rie \u00E9s convergent si la seq\u00FC\u00E8ncia de les seves sumes parcials tendeix a un l\u00EDmit; \u00E9s a dir que les sumes parcials s'acosten m\u00E9s i m\u00E9s a un determinat nombre quan el nombre de termes augmenta. M\u00E9s precisament, una s\u00E8rie convergeix si existeix un nombre tal que per qualsevol nombre positiu petit i arbitrari , existeix un enter suficientment gran tal que per tot , Si la s\u00E8ries \u00E9s convergent, el nombre (necess\u00E0riament \u00FAnic) s'anomena 'suma de la s\u00E8rie. Qualsevol s\u00E8rie no convergent s'anomena s\u00E8rie divergent."@ca . "\u0641\u064A \u0627\u0644\u0631\u064A\u0627\u0636\u064A\u0627\u062A\u060C \u0645\u062A\u0633\u0644\u0633\u0644\u0629 (\u0628\u0627\u0644\u0625\u0646\u062C\u0644\u064A\u0632\u064A\u0629: Convergent series)\u200F \u0647\u064A \u0645\u062C\u0645\u0648\u0639 \u062D\u062F\u0648\u062F \u0645\u062A\u062A\u0627\u0644\u064A\u0629 \u0645\u0646 \u0627\u0644\u0623\u0639\u062F\u0627\u062F. \u0644\u062A\u0643\u0646 \u0645\u062A\u062A\u0627\u0644\u064A\u0629 \u0645\u0627. \u0627\u0644\u062D\u062F \u0627\u0644\u0646\u0648\u0646\u064A \u0644\u0644\u0645\u062C\u0645\u0648\u0639 \u0627\u0644\u062C\u0632\u0626\u064A \u0647\u0648 \u0645\u062C\u0645\u0648\u0639 \u0627\u0644\u062D\u062F\u0648\u062F n \u0627\u0644\u0623\u0648\u0644\u0649 \u0644\u0644\u0645\u062A\u062A\u0627\u0644\u064A\u0629\u060C \u0623\u064A: \u062A\u0643\u0648\u0646 \u0645\u062A\u0633\u0644\u0633\u0644\u0629 \u0645\u0627 \u0645\u062A\u0642\u0627\u0631\u0628\u0629 \u0625\u0630\u0627 \u0643\u0627\u0646\u062A \u0645\u062A\u062A\u0627\u0644\u064A\u0629 \u0627\u0644\u0645\u062C\u0627\u0645\u064A\u0639 \u0627\u0644\u062C\u0632\u0626\u064A\u0629 \u0645\u062A\u0642\u0627\u0631\u0628\u0629. \u0648\u0628\u0634\u0643\u0644 \u0631\u0633\u0645\u064A\u060C \u062A\u0643\u0648\u0646 \u0645\u062A\u0633\u0644\u0633\u0644\u0629 \u0645\u062A\u0642\u0627\u0631\u0628\u0629 \u0625\u0630\u0627 \u0648\u064F\u062C\u062F\u062A \u0646\u0647\u0627\u064A\u0629 \u062D\u064A\u062B \u0643\u064A\u0641\u0645\u0627 \u0643\u0627\u0646 \u0639\u062F\u062F \u0645\u0648\u062C\u0628 \u0635\u063A\u064A\u0631 \u0645\u0627 \u060C \u0641\u0625\u0646\u0647 \u064A\u0648\u062C\u062F \u0639\u062F\u062F \u062D\u064A\u062B \u0645\u0647\u0645\u0627 \u0643\u0627\u0646 \u0641\u0625\u0646 : \u064A\u0642\u0627\u0644 \u0639\u0646 \u0645\u062A\u0633\u0644\u0633\u0644\u0629 \u063A\u064A\u0631 \u0645\u062A\u0642\u0627\u0631\u0628\u0629 \u0645\u062A\u0633\u0644\u0633\u0644\u0629 \u0645\u062A\u0628\u0627\u0639\u062F\u0629."@ar . "In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence defines a series S that is denoted The nth partial sum Sn is the sum of the first n terms of the sequence; that is, A series is convergent (or converges) if the sequence of its partial sums tends to a limit; that means that, when adding one after the other in the order given by the indices, one gets partial sums that become closer and closer to a given number. More precisely, a series converges, if there exists a number such that for every arbitrarily small positive number , there is a (sufficiently large) integer such that for all , If the series is convergent, the (necessarily unique) number is called the sum of the series. The same notation is used for the series, and, if it is convergent, to its sum. This convention is similar to that which is used for addition: a + b denotes the operation of adding a and b as well as the result of this addition, which is called the sum of a and b. Any series that is not convergent is said to be divergent or to diverge."@en . . . "Serie konbergentea bere gaien batura partzialen segidak limite finitua duen seriea da; kasu horretan, serieak batura finitua du, batura partzialen segidaren limitea hain zuzen."@eu . . . "In de wiskunde is convergentie een eigenschap van sommige rijen dat naarmate men verder in de rij komt de elementen van de rij een bepaalde waarde blijken te naderen. Zo'n rij heet convergent en de benaderde waarde wordt de limiet van de rij genoemd. De termen (getallen) in de rij heten convergenten. Zo convergeert de rij overduidelijk naar de limiet 0. In de wiskunde is men nog iets preciezer, en wordt de genoemde rij bijvoorbeeld niet als convergent beschouwd in de positieve getallen, omdat er geen positief getal is waarnaar de rij streeft. Men zegt dat een numerieke rij convergeert naar een bepaald getal L, de limiet van de rij genoemd, als geldt dat voor elke omgeving, hoe klein ook, van dat getal L, vanaf een bepaald element in de rij alle volgende elementen van de rij tot de gekozen omgeving behoren. Dit wordt precies geformuleerd in de volgende definitie."@nl . "Serie convergente"@es . "Serie convergente"@it . . . "11229"^^ . . "\uC218\uD559\uC5D0\uC11C \uAE09\uC218\uB780 \uC218\uC5F4\uC744 \uAD6C\uC131\uD558\uB294 \uD56D\uB4E4\uC744 \uD569\uC73C\uB85C \uB098\uD0C0\uB0B8 \uAC83\uC744 \uB9D0\uD55C\uB2E4. \uAE09\uC218\uC758 \uC218\uB834\uC5D0 \uAD00\uD55C \uB17C\uC758\uC5D0\uC11C \uAE09\uC218\uB294 \uBB34\uD55C\uAE09\uC218\uB97C \uB9D0\uD558\uBA70, \uC8FC\uC694 \uBB38\uC81C\uB294 \uC8FC\uC5B4\uC9C4 \uAE09\uC218\uC758 \uC218\uB834\uC5EC\uBD80\uC640 \uC218\uB834\uD560 \uACBD\uC6B0 \uADF8 \uD569\uC5D0 \uAD00\uD55C \uAC83\uC774\uB2E4. \uC218\uB834\uAE09\uC218\uB77C\uACE0 \uD574\uB3C4 \uADF8 \uD569\uC774 \uC54C\uB824\uC838 \uC788\uC9C0 \uC54A\uC740 \uACBD\uC6B0\uAC00 \uB9CE\uB2E4."@ko . . . . . . . . . "Convergent series"@en . "En math\u00E9matiques, une s\u00E9rie est dite convergente si la suite de ses sommes partielles a une limite dans l'espace consid\u00E9r\u00E9. Dans le cas contraire, elle est dite divergente. Pour des s\u00E9ries num\u00E9riques, ou \u00E0 valeurs dans un espace de Banach \u2014 c'est-\u00E0-dire un espace vectoriel norm\u00E9 complet \u2014, il suffit de prouver la convergence absolue de la s\u00E9rie pour montrer sa convergence, ce qui permet de se ramener \u00E0 une s\u00E9rie \u00E0 termes r\u00E9els positifs. Pour \u00E9tudier ces derni\u00E8res, il existe une large vari\u00E9t\u00E9 de r\u00E9sultats, tous fond\u00E9s sur le principe de comparaison."@fr . . . . "\uC218\uB834\uAE09\uC218"@ko . "\u6570\u5B66\u306B\u304A\u3044\u3066\u3001\u53CE\u675F\u7D1A\u6570 (convergent series) \u3068\u306F\u3001\u305D\u306E\u90E8\u5206\u548C\u306E\u6210\u3059\u6570\u5217\u304C\u53CE\u675F\u3059\u308B\u3088\u3046\u306A\u7D1A\u6570\u3067\u3042\u308B\u3002 \u3053\u3053\u3067\u3001\u7D1A\u6570\u3068\u306F\u6570\u5217\u306E\u9805\u306E\u7DCF\u548C\u306E\u3053\u3068\u3067\u3042\u308A\u3001\u4E0E\u3048\u3089\u308C\u305F\u6570\u5217 a1, a2, ..., an, ... \u306E\u7B2C n-\u90E8\u5206\u548C\u3068\u306F\u6700\u521D\u306E n-\u9805\u306E\u6709\u9650\u548C \u306E\u3053\u3068\u3092\u6307\u3059\u3002 \u3042\u308B\u7D1A\u6570\u304C\u53CE\u675F\u7D1A\u6570\u3067\u3042\u308B\u3053\u3068\u306F\u3001\u300C\uFF08\u6709\u9650\u306A\uFF09\u548C\u3092\u6301\u3064\u300D\u3068\u304B\u300C\u548C\u304C\u6709\u9650\u78BA\u5B9A\u3067\u3042\u308B\u300D\u306A\u3069\u3068\u8A00\u3044\u8868\u3055\u308C\u308B\u3002"@ja . . "\uC218\uD559\uC5D0\uC11C \uAE09\uC218\uB780 \uC218\uC5F4\uC744 \uAD6C\uC131\uD558\uB294 \uD56D\uB4E4\uC744 \uD569\uC73C\uB85C \uB098\uD0C0\uB0B8 \uAC83\uC744 \uB9D0\uD55C\uB2E4. \uAE09\uC218\uC758 \uC218\uB834\uC5D0 \uAD00\uD55C \uB17C\uC758\uC5D0\uC11C \uAE09\uC218\uB294 \uBB34\uD55C\uAE09\uC218\uB97C \uB9D0\uD558\uBA70, \uC8FC\uC694 \uBB38\uC81C\uB294 \uC8FC\uC5B4\uC9C4 \uAE09\uC218\uC758 \uC218\uB834\uC5EC\uBD80\uC640 \uC218\uB834\uD560 \uACBD\uC6B0 \uADF8 \uD569\uC5D0 \uAD00\uD55C \uAC83\uC774\uB2E4. \uC218\uB834\uAE09\uC218\uB77C\uACE0 \uD574\uB3C4 \uADF8 \uD569\uC774 \uC54C\uB824\uC838 \uC788\uC9C0 \uC54A\uC740 \uACBD\uC6B0\uAC00 \uB9CE\uB2E4."@ko . . "Serie konbergentea bere gaien batura partzialen segidak limite finitua duen seriea da; kasu horretan, serieak batura finitua du, batura partzialen segidaren limitea hain zuzen."@eu . "Converg\u00E8ncia (s\u00E8ries)"@ca . . . . . "In matematica, una serie convergente \u00E8 una serie tale che il limite delle sue somme parziali \u00E8 finito. Questo vuol dire che, data una successione , la serie \u00E8 convergente se la successione delle somme parziali ha un limite finito, cio\u00E8 se esiste finito tale che per ogni esiste tale che per ogni Il numero \u00E8 detto somma della serie: spesso \u00E8 difficile trovare questo numero, sebbene possa essere facile capire che una serie \u00E8 convergente."@it . "En matem\u00E1ticas, una serie (suma de los t\u00E9rminos de una secuencia de n\u00FAmeros), resulta convergente si la sucesi\u00F3n de sumas parciales tiene un l\u00EDmite en el espacio considerado. De otro modo, constituir\u00EDa lo que se denomina serie divergente."@es . . . . . . 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