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Continued fractions for finite sums. (English) Zbl 0829.11005

Given a finite sum \(S_n : = \sum^n_{i = 0} b_i\) with \(b_i \neq 0\) for all \(i > 0\), the author finds a continued fraction expansion \(a_0 + {\overset 2^n - {1} {\underset i = {1} K}} (1/a_i)\) such that \[ S_k = \sum^k_{i = 0} b_i = a_0 + {\overset 2^k - {1} {\underset i = {1} K}} (1/a_i) \quad \text{for } 0 \leq k \leq n. \] The continued fraction has the form
\[ \begin{aligned} S_0 & = a_0 = b_0 \\ S_2 & = a_0 + {1 \over a_1} \\ S_2 & = a_0 + K^3_{i = 1} (1/a_i) = a_0 + {1 \over a_1} + {1 \over a_2} + {1 \over - a_1} \\ S_3 & = a_0 + K^7_{i = 1} (1/a_i) = a_0 + {1 \over a_1} + {1 \over a_2} + {1 \over - a_1} + {1 \over a_4} + {1 \over a_1} + {1 \over -a_2} + {1 \over - a_1} \\ & = a_0 + {1 \over a_1} + {1 \over a_2} + {1 \over a_3} + {1 \over a_4} + {1 \over - a_3} + {1 \over - a_2} + {1 \over - a_1} \\ & \text{etc.} \end{aligned} \] (There are some misprints in Theorem 1.) The results can for instance be seen as corollaries to Euler’s continued fraction expansion of a given sum \(\sum^\infty_{i = 0} b_i\) [L. Euler, Introductio in Analysis Infinitorum, Vol. 1, Chapter 18 (1748)]. [See for instance, W. B. Jones and W. J. Thron, Continued fractions, Addision-Wesley (1980; Zbl 0445.30003), p. 37]).

MSC:

11A55 Continued fractions
40A15 Convergence and divergence of continued fractions

Citations:

Zbl 0445.30003
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References:

[1] Hardy, G.H. and Wright, E.M., An Introduction to the Theory of Numbers, Oxford University Press, 1979. · Zbl 0423.10001
[2] Shallit, J.O. , Simple Continued Fractions for Some Irrational Numbers , Journal of Number Theory , vol. 11 ( 1979) , p. 209-217. · Zbl 0404.10003
[3] Shallit, J.O. , Simple Continued Fractions for Some Irrational Numbers II, Journal of Number Theory , vol. 14 (1982), p. 228-231. · Zbl 0481.10005
[4] Thakur, D.S., Continued Fraction for the Exponential for Fq[t], Journal of Number Theory , vol. 41 (1992), p. 150-155. · Zbl 0754.11019
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