求函数项级数收敛区间的一种新方法

A New Method to Find the Convergent Interval of Functional Term Series

对形如∑∞n=0anxkn+b(k∈,b∈)的幂级数,当其缺项的时候,不能直接用公式ρ=li mn→∞an+1an求其收敛半径与收敛区间(本文约定收敛区间不含端点),一般都是直接采用达朗贝尔(比值)判别法求其收敛半径与收敛区间.事实上,对这种幂级数只需先作一个变量代换,就可以采用公式法求解.本文给出了这种方法的理论证明,并将结论进行了推广,即利用变量代换与公式法同样可求形如∑∞anxkn+bs(k,s∈,b∈)形式的函数项级数的收敛区间.

For series like ↑∞∑↓n=0 anx^kn＋b（k∈H,b∈Z） , it is usual to find the interval and radius of convergence by the d＇Alembert determination method, not to do those by the formula of ρ=lim↓n→∞｜（an＋1）/an｜ , when the series lacks terms. In fact, to find the interval and radius of convergence of those series by the formula of ρ=lim↓n→∞｜（an＋1）/an｜ , it is just sufficient to make a transformation. The proof of the method is shown in this paper. Besides, the method has been generalized to find the convergent intervals of functional term series like ↑∞∑↓n=0 anx^kn＋b（k∈H,b∈Z） by the formula of ρ=lim↓n→∞｜（an＋1）/an｜ with a transformation.