Some Fast Approximation Sequences Of The Euler Constant

The Euler constant γ is an important mathematical constant with a long history, and the study on it in all aspects has been enduring. This paper mainly discusses some approximation sequences of the Euler constant, using a variety of methods to construct some different approximation sequences and give the fast computation of the Euler constant. The paper is divided into five chapters.In Chapter 1,the basic concepts, the related properties and the research history of the Euler constant are introduced, meanwhile the current research on the sequences approximating towards the Euler constant is briefly described.In Chapter 2, further improvement on the approximation sequences of the Euler constant which were proposed by Vernescu [21] and Mortici [22]. We construct two classes of approximation sequences:odd fractional sequence and even fractional sequence. We can identify the parameters in the sequence and order of approximation by using the Bernoulli number and the lemma about the convergent sequence order. And also gives error bounds for several sequences. The rate of the approximation is faster than the sequences given by Vernescu and Mortici.In Chapter 3, based on the approximation sequences which were proposed by Lu [36], we construct two kinds of approximation sequences:sequence continued fraction containing n and sequence continued fraction containing n2. By using the lemma about the convergent sequence order, we can identify the parameters in the sequence and order of approximation. The rate of approximation is better than that of the sequences given by Lu.In Chapter 4, we determine a generalized sequence which converges to the Euler constant 7 with p+1 parameters based on ordinary Bell polynomial. The recursive relation between the parameters is obtained. Especially for p=7, the approximation sequence and asymptotic result are also given.In Chapter 5, we have carried on the summary to the full paper, and point out the idea and direction for further research.