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This book is the mathematical reference book which is helpful and accessible for not only professional mathematicians but also readers who have a little knowledge of mathematics.Mathematical constants have important applications in almost every field of engineering and science.In the study of mathematical constants, the remarkable trend is to find more accurate approximations and inequalities for them. Hence, many mathematicians and scientists have made great efforts in this subject of establishing more precise approximations for the mathematical constants and had lots of inspiring results. In this book, we extend our remarkable results and offer a more comprehensive overview.The book contains 3 chapters.Chapter 1 describes new sequences with continued fraction towards the Euler-Mascheroni constant and generalized Euler-Mascheroni constant, and summarizes several methods often employed to evolute the rate of convergence of our new sequences.Chapter 2 deals with our new classes ofsequences converging to the Lugo's constant based on continued fraction, and summarizes several methods often employed to evolute the rate of convergence of our new sequences.