大一求极限(统一n→∞)5.用数列极限的几何意义说明:liman=a的充要条件是lima(2n+1)=lima(2n)=a

Question

大一求极限
(统一n→∞)
5.用数列极限的几何意义说明:liman=a的充要条件是lima(2n+1)=lima(2n)=a

Answer 1

如果lima(2n)=a,lima(2n+1)=a,n=0时,lima(0)=a成立.令可k=2n ,则有lima(k)=a,lima(k+1)=a,根据数学归纳法,必有liman=a存在,lima(2n+1)=lima(2n)=a是liman=a的充分条件,
如果liman=a,当n为奇数时令n=2k+1,当n为偶数时令n=2k,则有lima(2k+1)=lima(2k)=a存在,lima(2n+1)=lima(2n)=a是liman=a的必要条件.