Research On The Historical Process Of Divergent Series

The history of the divergent series is an important part of the mathematics history. It is inextricablylinked with the development of calculus.It is widely used in other disciplines such as mathematics andphysical. Based on this,with the development of divergent series as the center,this paper systematicallyinspects the whole picture of divergent series twentieth century ago, mathematically reappears the workof great.The main findings are as follows:Firstly, we reviewed the early history of series operation, to show the diferent mathematicianshave done the work about series representation at diferent times. we find that before Cauchy it has noclear understanding of convergence and divergence. They also do not take convergence and divergenceseriously. Meanwhile,it caused confusion in understanding of series.Secondly, we have a more detailed discussion about harmonic series and alternating series by math-ematicians before the19th century. It pointed out that James Bernoulli proved harmonic series to theinfinite is instructive.John Bernoulli considered harmonic series as convergent series in the modern sensein the process of analysis. But most of the mathematicians take the alternating series summation as thefunction value of this series expressed.Thirdly, we detailedly study the work of Euler in the divergent series.we analysis its methods andsteps of Euler’s constant, numerical approximation,divergent summation and so on. I think that Euler hasa in-depth study of divergent series, it also have some inspiring views.He also concerned about how cleverto give the summation of divergent series. It pointed out that use the way of Euler can be established astrict way. The idea of series summation can still inspire people to consider its broad summation underthat circumstances.Last not but least, we have summarize that the mathematicians further research divergent series inCauchy sense, and analysis the application of divergent series. Meanwhile, sketching the historic con-tributions of mathematicians in sum property. It pointed out that divergent series have great significancein expressed function and in calculation. So mathematicians get asymptotic series, but it has diferent between gradual and convergence. Found that the theory of divergent series can be seen as a convergenceconcept to promote or expand. It uses the new methods to define the summation of divergent series inCauchy sense.We study the Poincare and others work,it pointed out that their discussed and results theyhave done in the summation of divergent series is laid a theoretical foundation for the study of divergentseries.