Some new approximations and inequalities of the sequence \((1+1/n)^n\) and improvements of Carleman’s inequality. (English) Zbl 1370.26005
Summary: In this paper, using the polynomial approximation and the continued fraction approximation, we present some sharp inequalities for the sequence \((1+1/n)^n\) and some applications to Carleman’s inequality. For demonstrating the superiority of our new inequalities over the classical one, some proofs and numerical computations are provided.
MSC:
| 26A09 | Elementary functions |
| 26D15 | Inequalities for sums, series and integrals |
| 11Y65 | Continued fraction calculations (number-theoretic aspects) |
Keywords:
polynomial approximation; continued fraction; constant e; Carleman inequality; double inequalitiesReferences:
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