幂级数逐项求导或积分后收敛半径不变的新证法

A New Method to Prove the Power Series with Unchanged Radius of Convergence after Itemized Derivative or Quadrature

要]运用数列极限的理论建立了关于数列上、下极限的相关命题,应用该命题和Cauchy-Hadamard定理的逆定理,给出了幂级数∞∑n=0anxn逐项求导、逐项积分后所得新的幂级数∞∑n=1nanxn-1和∞∑n=0ann+1xn+1收敛半径不变的性质的一个新的证明方法。该证明方法较传统的证明(基于Abel定理与正项级数的比较判别法)更为简洁。上述关于实幂级数结论的证明方法,可以推广到复幂级数上去。

In this paper,aproposition of superior and inferior limit is established with the theory of sequence limit.The proposition and Cauchy-Hadamard theorem are used a new method is provide to prove that the power series still have the same radius of convergence after itemized derivative or quadrature of method.And compared with the traditional method that is based on the theory of Abel and the comparison of positive series criterion,the new way is more simple.