. . "Formala potencoserio"@eo . . "\u5F62\u5F0F\u7684\u51AA\u7D1A\u6570"@ja . . . . . . . . . . . . . "En matem\u00E0tica, una s\u00E8rie formal de pot\u00E8ncies (de vegades s\u00E8rie de pot\u00E8ncies formal) \u00E9s una expressi\u00F3 matem\u00E0tica que est\u00E9n les propietats de les s\u00E8ries de pot\u00E8ncies en cossos com el dels reals o el dels complexos, permetent donar sentit formal a diverses notacions que t\u00E8cnicament no tenen rigor. Les s\u00E8ries formals de pot\u00E8ncies tenen diverses aplicacions, podent esmentar la combinat\u00F2ria i la teoria de nombres."@ca . . "Formale Potenzreihe"@de . . . . . . . . . "\u0424\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u0438\u0439 \u0441\u0442\u0435\u043F\u0435\u043D\u0435\u0432\u0438\u0439 \u0440\u044F\u0434"@uk . "In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, whose terms are of the form where is the th power of a variable ( is a non-negative integer), and is called the coefficient. Hence, power series can be viewed as a generalization of polynomials, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the are used only as position-holders for the coefficients, so that the coefficient of is the fifth term in the sequence. In combinatorics, the method of generating functions uses formal power series to represent numerical sequences and multisets, for instance allowing concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved. More generally, formal power series can include series with any finite (or countable) number of variables, and with coefficients in an arbitrary ring. Rings of formal power series are complete local rings, and this allows using calculus-like methods in the purely algebraic framework of algebraic geometry and commutative algebra. They are analogous in many ways to p-adic integers, which can be defined as formal series of the powers of p."@en . . "\u6570\u5B66\u306B\u304A\u3044\u3066\u3001\u5F62\u5F0F\u7684\u51AA\u7D1A\u6570\uFF08\u3051\u3044\u3057\u304D\u3066\u304D\u3079\u304D\u304D\u3085\u3046\u3059\u3046\u3001\u82F1: formal power series\uFF09\u3068\u306F\u3001\uFF08\u5F62\u5F0F\u7684\uFF09\u591A\u9805\u5F0F\u306E\u4E00\u822C\u5316\u3067\u3042\u308A\u3001\u591A\u9805\u5F0F\u304C\u6709\u9650\u500B\u306E\u9805\u3057\u304B\u6301\u305F\u306A\u3044\u306E\u306B\u5BFE\u3057\u3001\u5F62\u5F0F\u7684\u51AA\u7D1A\u6570\u306F\u9805\u304C\u6709\u9650\u500B\u3067\u306A\u304F\u3066\u3082\u3088\u3044\u3002\u4F8B\u3048\u3070\u3001\uFF08X \u3092\u4E0D\u5B9A\u5143\u3068\u3057\u3066\uFF09 \u306F\uFF08\u591A\u9805\u5F0F\u3067\u306F\u306A\u3044\uFF09\u51AA\u7D1A\u6570\u3067\u3042\u308B\u3002"@ja . . "51166"^^ . . . . "Serie formal de potencias"@es . . "Je algebro, formala potencoserio estas formala sumo de senfinaj termoj de potencoj de iu formala variablo, kiu ne devas konver\u011Di. La formalaj potencoserioj formas ringon, simile al la ringo de polinomoj."@eo . . "En matem\u00E1tica, se llama serie formal de potencias (a veces serie de potencias formal) a una expresi\u00F3n matem\u00E1tica que extiende las propiedades de las series de potencias en cuerpos como el de los reales o el de los complejos, permitiendo dar sentido formal a diversas notaciones que t\u00E9cnicamente carecen de rigurosidad. Las series formales de potencias tienen diversas aplicaciones, pudi\u00E9ndose mencionar la combinatoria y la teor\u00EDa de n\u00FAmeros."@es . . . "In matematica, le serie formali di potenze sono entit\u00E0 che rendono possibile riformulare gran parte dei risultati concernenti le serie di potenze ottenuti nella analisi matematica in ambiti formali che non si pongono questioni di \"convergenza\". Esse si rivelano utili, specialmente nella combinatoria, per fornire rappresentazioni compatte di successioni di numeri e funzioni e per ottenere formule chiuse per successioni definite attraverso un algoritmo ricorsivo; questo modo di operare viene detto metodo delle funzioni generatrici."@it . . "\uB300\uC218\uD559\uC5D0\uC11C \uD615\uC2DD\uC801 \uBA71\uAE09\uC218(\uC911\uAD6D\uC5B4: \u5F62\u5F0F\u7684\u51AA\u7D1A\u6578, \uC601\uC5B4: formal power series)\uB294 \uC218\uB834\uD560 \uD544\uC694\uAC00 \uC5C6\uB294 \uBA71\uAE09\uC218\uC774\uB2E4."@ko . "In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). Rings of formal power series are complete local rings, and this allows using calculus-like methods in the purely algebraic framework of algebraic geometry and commutative algebra. They are analogous in many ways to p-adic integers, which can be defined as formal series of the powers of p."@en . . . "\u0424\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u044B\u0439 \u0441\u0442\u0435\u043F\u0435\u043D\u043D\u043E\u0439 \u0440\u044F\u0434"@ru . "En matem\u00E0tica, una s\u00E8rie formal de pot\u00E8ncies (de vegades s\u00E8rie de pot\u00E8ncies formal) \u00E9s una expressi\u00F3 matem\u00E0tica que est\u00E9n les propietats de les s\u00E8ries de pot\u00E8ncies en cossos com el dels reals o el dels complexos, permetent donar sentit formal a diverses notacions que t\u00E8cnicament no tenen rigor. Les s\u00E8ries formals de pot\u00E8ncies tenen diverses aplicacions, podent esmentar la combinat\u00F2ria i la teoria de nombres."@ca . . . . . . . . . . . . . "Formal power series"@en . . . . . "\u6570\u5B66\u306B\u304A\u3044\u3066\u3001\u5F62\u5F0F\u7684\u51AA\u7D1A\u6570\uFF08\u3051\u3044\u3057\u304D\u3066\u304D\u3079\u304D\u304D\u3085\u3046\u3059\u3046\u3001\u82F1: formal power series\uFF09\u3068\u306F\u3001\uFF08\u5F62\u5F0F\u7684\uFF09\u591A\u9805\u5F0F\u306E\u4E00\u822C\u5316\u3067\u3042\u308A\u3001\u591A\u9805\u5F0F\u304C\u6709\u9650\u500B\u306E\u9805\u3057\u304B\u6301\u305F\u306A\u3044\u306E\u306B\u5BFE\u3057\u3001\u5F62\u5F0F\u7684\u51AA\u7D1A\u6570\u306F\u9805\u304C\u6709\u9650\u500B\u3067\u306A\u304F\u3066\u3082\u3088\u3044\u3002\u4F8B\u3048\u3070\u3001\uFF08X \u3092\u4E0D\u5B9A\u5143\u3068\u3057\u3066\uFF09 \u306F\uFF08\u591A\u9805\u5F0F\u3067\u306F\u306A\u3044\uFF09\u51AA\u7D1A\u6570\u3067\u3042\u308B\u3002"@ja . . . "\u5F62\u5F0F\u5E42\u7EA7\u6570(formal power series)\u662F\u4E00\u4E2A\u6570\u5B66\u4E2D\u7684\u62BD\u8C61\u6982\u5FF5\uFF0C\u662F\u4ECE\u5E42\u7EA7\u6570\u4E2D\u62BD\u79BB\u51FA\u6765\u7684\u4EE3\u6570\u5BF9\u8C61\u3002\u5F62\u5F0F\u5E42\u7EA7\u6570\u548C\u4ECE\u591A\u9879\u5F0F\u4E2D\u5265\u79BB\u51FA\u6765\u7684\u591A\u9879\u5F0F\u73AF\u7C7B\u4F3C\uFF0C\u4E0D\u8FC7\u5141\u8BB8\uFF08\u53EF\u6570\uFF09\u65E0\u7A77\u591A\u9879\u56E0\u5B50\u76F8\u52A0\uFF0C\u4F46\u4E0D\u50CF\u5E42\u7EA7\u6570\u4E00\u822C\u8981\u6C42\u7814\u7A76\u662F\u5426\u6536\u655B\u548C\u662F\u5426\u6709\u786E\u5B9A\u7684\u53D6\u503C\u3002\u5F62\u5F0F\u5E42\u7EA7\u6570\u5728\u4EE3\u6570\u548C\u7EC4\u5408\u7406\u8BBA\u4E2D\u6709\u5E7F\u6CDB\u5E94\u7528\u3002"@zh . . . "\u0424\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u044B\u0439 \u0441\u0442\u0435\u043F\u0435\u043D\u043D\u043E\u0301\u0439 \u0440\u044F\u0434 \u2014 \u0444\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u043E\u0435 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043A\u043E\u0435 \u0432\u044B\u0440\u0430\u0436\u0435\u043D\u0438\u0435 \u0432\u0438\u0434\u0430: \u0432 \u043A\u043E\u0442\u043E\u0440\u043E\u043C \u043A\u043E\u044D\u0444\u0444\u0438\u0446\u0438\u0435\u043D\u0442\u044B \u043F\u0440\u0438\u043D\u0430\u0434\u043B\u0435\u0436\u0430\u0442 \u043D\u0435\u043A\u043E\u0442\u043E\u0440\u043E\u043C\u0443 \u043A\u043E\u043B\u044C\u0446\u0443 . \u0412 \u043E\u0442\u043B\u0438\u0447\u0438\u0435 \u043E\u0442 \u0441\u0442\u0435\u043F\u0435\u043D\u043D\u044B\u0445 \u0440\u044F\u0434\u043E\u0432 \u0432 \u0430\u043D\u0430\u043B\u0438\u0437\u0435, \u0444\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u044B\u043C \u0441\u0442\u0435\u043F\u0435\u043D\u043D\u044B\u043C \u0440\u044F\u0434\u0430\u043C \u043D\u0435 \u043F\u0440\u0438\u0434\u0430\u0451\u0442\u0441\u044F \u0447\u0438\u0441\u043B\u043E\u0432\u044B\u0445 \u0437\u043D\u0430\u0447\u0435\u043D\u0438\u0439 \u0438 \u0441\u0445\u043E\u0434\u0438\u043C\u043E\u0441\u0442\u044C \u0442\u0430\u043A\u0438\u0445 \u0440\u044F\u0434\u043E\u0432 \u043D\u0435 \u0440\u0430\u0441\u0441\u043C\u0430\u0442\u0440\u0438\u0432\u0430\u0435\u0442\u0441\u044F. \u0424\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u044B\u0435 \u0441\u0442\u0435\u043F\u0435\u043D\u043D\u044B\u0435 \u0440\u044F\u0434\u044B \u0438\u0441\u0441\u043B\u0435\u0434\u0443\u044E\u0442\u0441\u044F \u0432 \u0430\u043B\u0433\u0435\u0431\u0440\u0435, \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0438\u0438, \u043A\u043E\u043C\u0431\u0438\u043D\u0430\u0442\u043E\u0440\u0438\u043A\u0435. \u041A\u0440\u043E\u043C\u0435 \u0442\u043E\u0433\u043E, \u043E\u043D\u0438 \u044F\u0432\u043B\u044F\u044E\u0442\u0441\u044F \u0443\u0434\u043E\u0431\u043D\u044B\u043C \u0438\u043D\u0441\u0442\u0440\u0443\u043C\u0435\u043D\u0442\u043E\u043C \u043F\u0440\u0438 \u0438\u0441\u0441\u043B\u0435\u0434\u043E\u0432\u0430\u043D\u0438\u0438 \u0440\u0430\u0437\u043B\u0438\u0447\u043D\u044B\u0445 \u0433\u043B\u0430\u0434\u043A\u0438\u0445 \u043E\u0431\u044A\u0435\u043A\u0442\u043E\u0432, \u043D\u0430\u043F\u0440\u0438\u043C\u0435\u0440, \u0432 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u043E\u0439 \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0438\u0438 \u0438 \u0442\u0435\u043E\u0440\u0438\u0438 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u044B\u0445 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0439."@ru . . . . . . . . "\uD615\uC2DD\uC801 \uBA71\uAE09\uC218"@ko . . . . . "In matematica, le serie formali di potenze sono entit\u00E0 che rendono possibile riformulare gran parte dei risultati concernenti le serie di potenze ottenuti nella analisi matematica in ambiti formali che non si pongono questioni di \"convergenza\". Esse si rivelano utili, specialmente nella combinatoria, per fornire rappresentazioni compatte di successioni di numeri e funzioni e per ottenere formule chiuse per successioni definite attraverso un algoritmo ricorsivo; questo modo di operare viene detto metodo delle funzioni generatrici."@it . . . . . . . "\u5F62\u5F0F\u5E42\u7EA7\u6570(formal power series)\u662F\u4E00\u4E2A\u6570\u5B66\u4E2D\u7684\u62BD\u8C61\u6982\u5FF5\uFF0C\u662F\u4ECE\u5E42\u7EA7\u6570\u4E2D\u62BD\u79BB\u51FA\u6765\u7684\u4EE3\u6570\u5BF9\u8C61\u3002\u5F62\u5F0F\u5E42\u7EA7\u6570\u548C\u4ECE\u591A\u9879\u5F0F\u4E2D\u5265\u79BB\u51FA\u6765\u7684\u591A\u9879\u5F0F\u73AF\u7C7B\u4F3C\uFF0C\u4E0D\u8FC7\u5141\u8BB8\uFF08\u53EF\u6570\uFF09\u65E0\u7A77\u591A\u9879\u56E0\u5B50\u76F8\u52A0\uFF0C\u4F46\u4E0D\u50CF\u5E42\u7EA7\u6570\u4E00\u822C\u8981\u6C42\u7814\u7A76\u662F\u5426\u6536\u655B\u548C\u662F\u5426\u6709\u786E\u5B9A\u7684\u53D6\u503C\u3002\u5F62\u5F0F\u5E42\u7EA7\u6570\u5728\u4EE3\u6570\u548C\u7EC4\u5408\u7406\u8BBA\u4E2D\u6709\u5E7F\u6CDB\u5E94\u7528\u3002"@zh . . . . . "\u0424\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u0438\u0439 \u0441\u0442\u0435\u043F\u0435\u043D\u0435\u0432\u0438\u0439 \u0440\u044F\u0434 \u2014 \u0444\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u0438\u0439 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0457\u0447\u043D\u0438\u0439 \u0432\u0438\u0440\u0430\u0437 \u0432\u0438\u0434\u0443: \u0432 \u044F\u043A\u043E\u043C\u0443 \u043A\u043E\u0435\u0444\u0456\u0446\u0456\u0454\u043D\u0442\u0438 \u043D\u0430\u043B\u0435\u0436\u0430\u0442\u044C \u0434\u0435\u044F\u043A\u043E\u043C\u0443 \u043A\u0456\u043B\u044C\u0446\u044E . \u041D\u0430 \u0432\u0456\u0434\u043C\u0456\u043D\u0443 \u0432\u0456\u0434 \u0441\u0442\u0435\u043F\u0435\u043D\u0435\u0432\u0438\u0445 \u0440\u044F\u0434\u0456\u0432 \u0443 \u0430\u043D\u0430\u043B\u0456\u0437\u0456 \u0444\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u0438\u043C \u0441\u0442\u0435\u043F\u0435\u043D\u0435\u0432\u0438\u043C \u0440\u044F\u0434\u0430\u043C \u043D\u0435 \u043D\u0430\u0434\u0430\u0454\u0442\u044C\u0441\u044F \u0447\u0438\u0441\u043B\u043E\u0432\u0438\u0445 \u0437\u043D\u0430\u0447\u0435\u043D\u044C \u0456 \u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u043D\u043E \u043D\u0435 \u043C\u0430\u0454 \u0437\u043C\u0456\u0441\u0442\u0443 \u0437\u0431\u0456\u0436\u043D\u0456\u0441\u0442\u044C \u0442\u0430\u043A\u0438\u0445 \u0440\u044F\u0434\u0456\u0432 \u0434\u043B\u044F \u0447\u0438\u0441\u043B\u043E\u0432\u0438\u0445 \u0430\u0440\u0433\u0443\u043C\u0435\u043D\u0442\u0456\u0432.\u0424\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u0456 \u0441\u0442\u0435\u043F\u0435\u043D\u0435\u0432\u0456 \u0440\u044F\u0434\u0438 \u0434\u043E\u0441\u043B\u0456\u0434\u0436\u0443\u044E\u0442\u044C\u0441\u044F \u0443 \u0430\u043B\u0433\u0435\u0431\u0440\u0456, \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0456\u0457, \u043A\u043E\u043C\u0431\u0456\u043D\u0430\u0442\u043E\u0440\u0438\u0446\u0456."@uk . . . . . . "Serie formale di potenze"@it . . . . . . . . . . . . "En matem\u00E1tica, se llama serie formal de potencias (a veces serie de potencias formal) a una expresi\u00F3n matem\u00E1tica que extiende las propiedades de las series de potencias en cuerpos como el de los reales o el de los complejos, permitiendo dar sentido formal a diversas notaciones que t\u00E9cnicamente carecen de rigurosidad. Las series formales de potencias tienen diversas aplicaciones, pudi\u00E9ndose mencionar la combinatoria y la teor\u00EDa de n\u00FAmeros."@es . "\u0424\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u0438\u0439 \u0441\u0442\u0435\u043F\u0435\u043D\u0435\u0432\u0438\u0439 \u0440\u044F\u0434 \u2014 \u0444\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u0438\u0439 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0457\u0447\u043D\u0438\u0439 \u0432\u0438\u0440\u0430\u0437 \u0432\u0438\u0434\u0443: \u0432 \u044F\u043A\u043E\u043C\u0443 \u043A\u043E\u0435\u0444\u0456\u0446\u0456\u0454\u043D\u0442\u0438 \u043D\u0430\u043B\u0435\u0436\u0430\u0442\u044C \u0434\u0435\u044F\u043A\u043E\u043C\u0443 \u043A\u0456\u043B\u044C\u0446\u044E . \u041D\u0430 \u0432\u0456\u0434\u043C\u0456\u043D\u0443 \u0432\u0456\u0434 \u0441\u0442\u0435\u043F\u0435\u043D\u0435\u0432\u0438\u0445 \u0440\u044F\u0434\u0456\u0432 \u0443 \u0430\u043D\u0430\u043B\u0456\u0437\u0456 \u0444\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u0438\u043C \u0441\u0442\u0435\u043F\u0435\u043D\u0435\u0432\u0438\u043C \u0440\u044F\u0434\u0430\u043C \u043D\u0435 \u043D\u0430\u0434\u0430\u0454\u0442\u044C\u0441\u044F \u0447\u0438\u0441\u043B\u043E\u0432\u0438\u0445 \u0437\u043D\u0430\u0447\u0435\u043D\u044C \u0456 \u0432\u0456\u0434\u043F\u043E\u0432\u0456\u0434\u043D\u043E \u043D\u0435 \u043C\u0430\u0454 \u0437\u043C\u0456\u0441\u0442\u0443 \u0437\u0431\u0456\u0436\u043D\u0456\u0441\u0442\u044C \u0442\u0430\u043A\u0438\u0445 \u0440\u044F\u0434\u0456\u0432 \u0434\u043B\u044F \u0447\u0438\u0441\u043B\u043E\u0432\u0438\u0445 \u0430\u0440\u0433\u0443\u043C\u0435\u043D\u0442\u0456\u0432.\u0424\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u0456 \u0441\u0442\u0435\u043F\u0435\u043D\u0435\u0432\u0456 \u0440\u044F\u0434\u0438 \u0434\u043E\u0441\u043B\u0456\u0434\u0436\u0443\u044E\u0442\u044C\u0441\u044F \u0443 \u0430\u043B\u0433\u0435\u0431\u0440\u0456, \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0456\u0457, \u043A\u043E\u043C\u0431\u0456\u043D\u0430\u0442\u043E\u0440\u0438\u0446\u0456."@uk . . . . "Die formalen Potenzreihen in der Mathematik sind eine Verallgemeinerung der Polynome der Polynomringe. Wie bei letzteren stehen bei ihnen die ringtheoretischen Eigenschaften im Vordergrund, w\u00E4hrend bei den Potenzreihen der Analysis der Schwerpunkt auf den analytischen, den (Grenzwert-)Eigenschaften, liegt. Gemeinsam ist, dass die Koeffizienten aus einem Ring genommen werden, der hier sehr beliebig sein kann, wogegen er in der Analysis ausschlie\u00DFlich ein vollst\u00E4ndiger Ring ist, meist der K\u00F6rper der reellen oder der komplexen Zahlen.Ein anderer Unterschied ist, dass die \u201EVariable\u201C eine Unbestimmte ist, die oft mit Gro\u00DFbuchstaben (oder ) notiert und der in der formalen Potenzreihe ein \u201EWert\u201C nicht zugewiesen wird.Die im Nullpunkt analytischen Potenzreihen der Analysis k\u00F6nnen auch als formale Potenzreihen aufgefasst werden, da sie wie diese beliebig oft sind und dem Koeffizientenvergleich unterliegen. Wegen der vielen gemeinsamen Eigenschaften und Begriffsbildungen werden die formalen Laurent-Reihen in diesem Artikel mitbehandelt.Die Definitionen und Eigenschaften sind bei den formalen Laurent-Reihen geringf\u00FCgig komplexer, enthalten aber sehr h\u00E4ufig die formalen Potenzreihen als Spezialfall. Unterst\u00FCtzung f\u00FCr das Rechnen mit formalen Potenz- und Laurent-Reihen gibt es in vielen Computeralgebra-Systemen."@de . . . . . . "Je algebro, formala potencoserio estas formala sumo de senfinaj termoj de potencoj de iu formala variablo, kiu ne devas konver\u011Di. La formalaj potencoserioj formas ringon, simile al la ringo de polinomoj."@eo . . . . . . . . . . . . . "S\u00E9rie formelle"@fr . . . . . "\u0424\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u044B\u0439 \u0441\u0442\u0435\u043F\u0435\u043D\u043D\u043E\u0301\u0439 \u0440\u044F\u0434 \u2014 \u0444\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u043E\u0435 \u0430\u043B\u0433\u0435\u0431\u0440\u0430\u0438\u0447\u0435\u0441\u043A\u043E\u0435 \u0432\u044B\u0440\u0430\u0436\u0435\u043D\u0438\u0435 \u0432\u0438\u0434\u0430: \u0432 \u043A\u043E\u0442\u043E\u0440\u043E\u043C \u043A\u043E\u044D\u0444\u0444\u0438\u0446\u0438\u0435\u043D\u0442\u044B \u043F\u0440\u0438\u043D\u0430\u0434\u043B\u0435\u0436\u0430\u0442 \u043D\u0435\u043A\u043E\u0442\u043E\u0440\u043E\u043C\u0443 \u043A\u043E\u043B\u044C\u0446\u0443 . \u0412 \u043E\u0442\u043B\u0438\u0447\u0438\u0435 \u043E\u0442 \u0441\u0442\u0435\u043F\u0435\u043D\u043D\u044B\u0445 \u0440\u044F\u0434\u043E\u0432 \u0432 \u0430\u043D\u0430\u043B\u0438\u0437\u0435, \u0444\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u044B\u043C \u0441\u0442\u0435\u043F\u0435\u043D\u043D\u044B\u043C \u0440\u044F\u0434\u0430\u043C \u043D\u0435 \u043F\u0440\u0438\u0434\u0430\u0451\u0442\u0441\u044F \u0447\u0438\u0441\u043B\u043E\u0432\u044B\u0445 \u0437\u043D\u0430\u0447\u0435\u043D\u0438\u0439 \u0438 \u0441\u0445\u043E\u0434\u0438\u043C\u043E\u0441\u0442\u044C \u0442\u0430\u043A\u0438\u0445 \u0440\u044F\u0434\u043E\u0432 \u043D\u0435 \u0440\u0430\u0441\u0441\u043C\u0430\u0442\u0440\u0438\u0432\u0430\u0435\u0442\u0441\u044F. \u0424\u043E\u0440\u043C\u0430\u043B\u044C\u043D\u044B\u0435 \u0441\u0442\u0435\u043F\u0435\u043D\u043D\u044B\u0435 \u0440\u044F\u0434\u044B \u0438\u0441\u0441\u043B\u0435\u0434\u0443\u044E\u0442\u0441\u044F \u0432 \u0430\u043B\u0433\u0435\u0431\u0440\u0435, \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0438\u0438, \u043A\u043E\u043C\u0431\u0438\u043D\u0430\u0442\u043E\u0440\u0438\u043A\u0435. \u041A\u0440\u043E\u043C\u0435 \u0442\u043E\u0433\u043E, \u043E\u043D\u0438 \u044F\u0432\u043B\u044F\u044E\u0442\u0441\u044F \u0443\u0434\u043E\u0431\u043D\u044B\u043C \u0438\u043D\u0441\u0442\u0440\u0443\u043C\u0435\u043D\u0442\u043E\u043C \u043F\u0440\u0438 \u0438\u0441\u0441\u043B\u0435\u0434\u043E\u0432\u0430\u043D\u0438\u0438 \u0440\u0430\u0437\u043B\u0438\u0447\u043D\u044B\u0445 \u0433\u043B\u0430\u0434\u043A\u0438\u0445 \u043E\u0431\u044A\u0435\u043A\u0442\u043E\u0432, \u043D\u0430\u043F\u0440\u0438\u043C\u0435\u0440, \u0432 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u043E\u0439 \u0442\u043E\u043F\u043E\u043B\u043E\u0433\u0438\u0438 \u0438 \u0442\u0435\u043E\u0440\u0438\u0438 \u0434\u0438\u0444\u0444\u0435\u0440\u0435\u043D\u0446\u0438\u0430\u043B\u044C\u043D\u044B\u0445 \u0443\u0440\u0430\u0432\u043D\u0435\u043D\u0438\u0439."@ru . . . "Die formalen Potenzreihen in der Mathematik sind eine Verallgemeinerung der Polynome der Polynomringe. Wie bei letzteren stehen bei ihnen die ringtheoretischen Eigenschaften im Vordergrund, w\u00E4hrend bei den Potenzreihen der Analysis der Schwerpunkt auf den analytischen, den (Grenzwert-)Eigenschaften, liegt. Unterst\u00FCtzung f\u00FCr das Rechnen mit formalen Potenz- und Laurent-Reihen gibt es in vielen Computeralgebra-Systemen."@de . . . . . . . . . . . . . . . . . . "S\u00E8rie formal de pot\u00E8ncies"@ca . . . "1093700634"^^ . . . "En alg\u00E8bre, les s\u00E9ries formelles sont une g\u00E9n\u00E9ralisation des polyn\u00F4mes autorisant des sommes infinies, de la m\u00EAme fa\u00E7on qu'en analyse, les s\u00E9ries enti\u00E8res g\u00E9n\u00E9ralisent les fonctions polynomiales, \u00E0 ceci pr\u00E8s que dans le cadre alg\u00E9brique, les probl\u00E8mes de convergence sont \u00E9vit\u00E9s par des d\u00E9finitions ad hoc. Ces objets sont utiles pour d\u00E9crire de fa\u00E7on concise des suites et pour trouver des formules pour des suites d\u00E9finies par r\u00E9currence via ce que l'on appelle les s\u00E9ries g\u00E9n\u00E9ratrices."@fr . . . "En alg\u00E8bre, les s\u00E9ries formelles sont une g\u00E9n\u00E9ralisation des polyn\u00F4mes autorisant des sommes infinies, de la m\u00EAme fa\u00E7on qu'en analyse, les s\u00E9ries enti\u00E8res g\u00E9n\u00E9ralisent les fonctions polynomiales, \u00E0 ceci pr\u00E8s que dans le cadre alg\u00E9brique, les probl\u00E8mes de convergence sont \u00E9vit\u00E9s par des d\u00E9finitions ad hoc. Ces objets sont utiles pour d\u00E9crire de fa\u00E7on concise des suites et pour trouver des formules pour des suites d\u00E9finies par r\u00E9currence via ce que l'on appelle les s\u00E9ries g\u00E9n\u00E9ratrices."@fr . . . . . . . . . . . . . . . . "\uB300\uC218\uD559\uC5D0\uC11C \uD615\uC2DD\uC801 \uBA71\uAE09\uC218(\uC911\uAD6D\uC5B4: \u5F62\u5F0F\u7684\u51AA\u7D1A\u6578, \uC601\uC5B4: formal power series)\uB294 \uC218\uB834\uD560 \uD544\uC694\uAC00 \uC5C6\uB294 \uBA71\uAE09\uC218\uC774\uB2E4."@ko . . . . . "\u5F62\u5F0F\u5E42\u7EA7\u6570"@zh . . . . . . "60012"^^ . . . .