数列[(-1)^n+1][(n+1)/n]的极限

题目

数列[(-1)^n+1][(n+1)/n]的极限
分n=2N和n=2N+1来求解?

答案

n=2N时,[(-1)^n + 1][(n+1)/n] = [(-1)^(2N) + 1]/[(2N+1)/(2N)] = (2N+1)/N = 2 + 1/N,
n=2N->无穷大时,N->无穷大,[(-1)^n + 1][(n+1)/n] = 2 + 1/N -> 2
n=2N+1时,[(-1)^n + 1][(n+1)/n] = [(-1)^(2N+1) + 1][(2N+2)/(2N+1)] = 0.
n=2N+1->无穷大时,[(-1)^n+1][(n+1)/n] -> 0 不等于 2.
因此,n->无穷大时,[(-1)^n+1][(n+1)/n]的极限不存在.