Abstract
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.
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Computations made in this paper were performed using Maple software.
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The research of the first author was supported by the National Natural Science Foundation of China (Grant No. 11571058) and the Fundamental Research Funds for the Central Universities (Grant No. DUT15LK19). The second author was supported by the National Natural Science Foundation of China (Grant No. 11371077).
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Lu, D., Song, L. & Liu, Z. Some new approximations and inequalities of the sequence \((1+1/n)^n\) and improvements of Carleman’s inequality. Ramanujan J 43, 69–82 (2017). https://doi.org/10.1007/s11139-015-9765-x
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DOI: https://doi.org/10.1007/s11139-015-9765-x