• Title/Summary/Keyword: Sum of Geometric Series

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묵사집산법의 수열

  • Heo, Min
    • Journal for History of Mathematics
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    • v.17 no.1
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    • pp.15-32
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    • 2004
  • In this article we survey the sequences and the series in Mooksajipsanbup(默思集算法) which is the seventeenth century mathematics book of Chosun dynasty. First, we classify them into three categories: arithmetics, geometric, and general sequences (series). And then we explore the old methods to find the values of terms and the sum of terms.

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The Metaphorical Model of Archimedes' Idea on the Sum of Geometrical Series (무한 등비급수의 합에 대한 Archimedes의 아이디어의 은유적 모델과 그 교육적 활용)

  • Lee, Seoung Woo
    • School Mathematics
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    • v.18 no.1
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    • pp.215-229
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    • 2016
  • This study aims to identify Archimedes' idea used while proving proposition 23 in 'Quadrature of the Parabola' and to provide an alternative way for finding the sum of geometric series without applying the concept of limit by extending the idea though metaphor. This metaphorical model is characterized as static and thus can be complimentary to the dynamic aspect of limit concept adopted in Korean high school mathematics textbooks. In addition, middle school students can understand $0.999{\cdots}=1$ with this model in a structural way differently from the operative one suggested in Korean middle school mathematics textbooks. In this respect, I argue that the metaphorical model can be an useful educational tool for Korean secondary students to overcome epistemological obstacles inherent in the concepts of infinity and limit by making it possible to transfer from geometrical context to algebraic context.

THE ZEROS OF CERTAIN FAMILY OF SELF-RECIPROCAL POLYNOMIALS

  • Kim, Seon-Hong
    • Bulletin of the Korean Mathematical Society
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    • v.44 no.3
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    • pp.461-473
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    • 2007
  • For integral self-reciprocal polynomials P(z) and Q(z) with all zeros lying on the unit circle, does there exist integral self-reciprocal polynomial $G_r(z)$ depending on r such that for any r, $0{\leq}r{\leq}1$, all zeros of $G_r(z)$ lie on the unit circle and $G_0(z)$ = P(z), $G_1(z)$ = Q(z)? We study this question by providing examples. An example answers some interesting questions. Another example relates to the study of convex combination of two polynomials. From this example, we deduce the study of the sum of certain two products of finite geometric series.

Practical Validity of Weighting Methods : A Comparative Analysis Using Bootstrapping (부트스트랩핑을 이용한 가중치 결정방법의 실질적 타당성 비교)

  • Jeong, Ji-Ahn;Cho, Sung-Ku
    • Journal of Korean Institute of Industrial Engineers
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    • v.26 no.1
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    • pp.27-35
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    • 2000
  • For a weighting method to be practically valid, it should produce weights which coincide with the relative importance of attributes perceived by the decision maker. In this paper, 'bootstrapping' is used to compare the practical validities of five weighting methods frequently used; the rank order centroid method, the rank reciprocal method, the rank sum method, the entropic method, and the geometric mean method. Bootstrapping refers to the procedure where the analysts allow the decision maker to make careful judgements on a series of similar cases, then infer statistically what weights he was implicitly using to arrive at the particular ranking. The weights produced by bootstrapping can therefore be regarded as well reflecting the decision maker's perceived relative importances. Bootstrapping and the five weighting methods were applied to a job selection problem. The results showed that both the rank order centroid method and the rank reciprocal method had higher level of practical validity than the other three methods, though a large difference could not be found either in the resulting weights or in the corresponding solutions.

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