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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 $$2$$.

##### MSC:
 11J82 Measures of irrationality and of transcendence 41A21 Padé approximation
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##### References:
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