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:''For an elementary calculus-based introduction, see Divergent series on Wikiversity'' In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series must approach zero. Thus any series in which the individual terms do not approach zero diverges. However, convergence is a stronger condition: not all series whose terms approach zero converge. A counterexample is the harmonic series : The divergence of the harmonic series was proven by the medieval mathematician Nicole Oresme. In specialized mathematical contexts, values can be objectively assigned to certain series whose sequence of partial sums diverges, this is to make meaning of the divergence of the series. A summability method or summation method is a partial function from the set of series to values. For example, Cesàro summation assigns Grandi's divergent series : the value 1/2. Cesàro summation is an averaging method, in that it relies on the arithmetic mean of the sequence of partial sums. Other methods involve analytic continuations of related series. In physics, there are a wide variety of summability methods; these are discussed in greater detail in the article on regularization. ==History== Before the 19th century divergent series were widely used by Euler and others, but often led to confusing and contradictory results. A major problem was Euler's idea that any divergent series should have a natural sum, without first defining what is meant by the sum of a divergent series. Cauchy eventually gave a rigorous definition of the sum of a (convergent) series, and for some time after this divergent series were mostly excluded from mathematics. They reappeared in 1886 with Poincaré's work on asymptotic series. In 1890 Cesaro realized that one could give a rigorous definition of the sum of some divergent series, and defined Cesaro summation. (This was not the first use of Cesaro summation which was used implicitly by Frobenius in 1880; Cesaro's key contribution was not the discovery of this method but his idea that one should give an explicit definition of the sum of a divergent series.) In the years after Cesaro's paper several other mathematicians gave other definitions of the sum of a divergent series, though these are not always compatible: different definitions can give different answers for the sum of the same divergent series, so when talking about the sum of a divergent series it is necessary to specify which summation method one is using. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Divergent series」の詳細全文を読む スポンサード リンク
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