求当n→∞时，[(2n+3)⼀(2n+1)]^(n+1)的极限

2025-06-28 04:07:00

推荐回答（3个）

回答1：

lim[(2n+3)/(2n+1)]^(n+1)
n→∞

=lim[(2n+1+2)/(2n+1)]^(n+1)
n→∞

=lim[1+2/(2n+1)]^(n+1)
n→∞

=lim[1+2/(2n+1)]^{[(2n+1)/2]2(n+1)/(2n+1)}

n→∞

=Ⅰime^2(n+1)/(2n+1)
n→∞

=e

回答2：

图

回答3：

lim [(2n+3)/(2n+1)]^(n+1)=lim {[1+2/(2n+1)]^(2n+1)/2}^[2(n+1)/(2n+1)]={lim [1+2/(2n+1)]^(2n+1)/2} ^[lim 2(n+1)/(2n+1)]=e^1=e