Question

设[∞∑ n=1] (an x^n)的收敛半径为R, 则[∞∑ n=1] [an x^(2n+1)]的收敛半径为

答案是：√R

Answer 1

lim |an+1x^(2n+3)|/|anx^(2n+1)|
=lim |an+1/an|x²=Rx²<1,
|x|<1/√R 收敛半径为√R