Study On Euler's Theory Of The Series

Infinite series has substantive developed since the creation of the calculus. In the 18th century, the formalized view took a dominate place in the theory of infinite series. During this period, some special types of series emerged, such as the asymptotic series, the recurrent series, the continued fractions and so on. The new types of series were beyond the original theory and lead the mathematicians a more formalized treatment of them. Euler, as a representitive of them, genaralized the early series theory by the formalization approach from many aspects. Euler developed a more formal approach which generalized the early theory. He transformed the infinite series from a ordinary calculation tool into an important subject and deepened the application of series to a new level, which laid the foundation for the future development of the series theory.This paper discusses Euler's original works on theory of the series, i.e. the harmonic series, the Basel problem, the Zeta function, the summation of divergent series, the numerical approximation, the solution of differential equations, the trigonometric series and the continued fractions. Based on the careful and thorough study of Euler's brilliant methods and ideas, the paper concludes that Euler's key method when dealing with the series theory is formalization and that the core of Euler's whole work in this field is the summation of the series.