Sum range of a quaternion series. (English. Russian original) Zbl 1365.40001
J. Math. Sci., New York 216, No. 4, 519-521 (2016); translation from Sovrem. Mat. Prilozh. 94 (2014).
Summary: In this paper, we obtain a result which implies, in particular, that for a quaternion \(z\notin \{-1,1\}\) with \(|z| = 1\), the sum range of the series \(\displaystyle \sum_n\frac{z^n}{n}\) is a closed proper subfield of the division ring of quaternions \(\mathbb{H}\) isometrically isomorphic to the field of complex numbers \(\mathbb{C}\).
MSC:
| 40A05 | Convergence and divergence of series and sequences |
References:
| [1] | G. Giorgobiani and V. Tarieladze, “On complex universal series,” Proc. A. Razmadze Math. Inst., 160, 53-63 (2012). · Zbl 1288.40001 |
| [2] | G. Giorgobiani and V. Tarieladze, “Special universal series, I,” in: Urgent Problems of Applied Mathematics and Mechanics, Nova Science Publishers (2013), pp. 125-130. |
| [3] | H. Hadwiger, “Eine Bemerkung über Umordnung von Reihen reeller Funktionen,” Tôhoku Math. J., 46, 22-25 (1939). · JFM 65.0226.01 |
| [4] | D. O. Shklyarskii, “Conditionally convergent series of vectors,” Usp. Mat. Nauk, 10, 51-59 (1944). · Zbl 0061.11807 |
| [5] | E. Steinitz, “Bedingt convergente Reichen konvexe Systeme. Teil I,” J. Math., 143, 128-175 (1913). · JFM 44.0287.01 |
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