2022-08-18T19:16:38Zhttps://kitami-it.repo.nii.ac.jp/oai
oai:kitami-it.repo.nii.ac.jp:000061842021-03-01T06:56:32Z
On Convergence of Infinite Continued Fractions無限連分数の収束について礒部, 煕郎31695application/pdfLet {ν_k} be a sequence of natural numbers. The following form （1） of the sequence {ν_k} is called a infinite continued fraction. 1/(ν_1+1)/(ν_2+1)/(…) (1) From the sequence {ν_k} j the sequence ｛ξ_k｝is made as follows　(ξ_k=1)/(ν_1+1)/(…)/1/(ν_k-1+1)/(ν-1) (2) and it is convergent sequence. In this paper, we give the following relationsξ_2＜ξ_4＜…＜<ξ_3＜ξ_1 (9) │ξ-ｋ-ξ-ｋ1│＜__-1/(k(k-1) (8) The conclusion drawn from these relations is the uniformity about convergence of infinite continued fractions. Namely, for any positive number s and sequence of natural numbers ε and sequence of natural numbers {ν_k}, there exists some natural number N such that N≦k, l implies │ξ-ｋ-ξ-1│＜εFurthermore, let S be the totality of all sequences of natural numbers and R be the totality of all irrational numbers in the open interval （0，I）. We define the metric function d in the space S as follows d(a, b)=1/(Min{k:ν_k≠μ_k} (S∋a, b, a={ν_k}, b={μ_k}). Ｔｈｅｓｐａｃｅ R is the subspace of the real line. In this paper, the relations between the spaces S and R are discussed.departmental bulletin paper北見工業大学1966-03application/pdf北見工業大学研究報告41109117https://kitami-it.repo.nii.ac.jp/record/6184/files/1-4-12.pdfjpn