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On the rational approximation of the sum of the reciprocals of the Fermat numbers. (English) Zbl 1271.11075
Ramanujan J. 30, No. 1, 39-65 (2013); addendum ibid. 37, No. 1, 109–111 (2015).
The author proves that if \(a\geq 2\) is a positive integer then the irrationality exponent of the numbers \[ \sum_{n=1}^\infty \frac{a^{2^n}}{1+a^{2^n}}\quad \text{and}\quad \sum_{n=1}^\infty \frac{a^{2^n}}{1-a^{2^n}} \] is equal to \(2\). The proof is based on the Padé approximation and uses the Hankel determinants. As a special case we obtain that the sum of reciprocal Fermat numbers has an irrationality exponent equal to \(2\).

11J82 Measures of irrationality and of transcendence
41A21 Padé approximation
Full Text: DOI
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