求教一道二项式定理的题目求证:2＜＝(1+1/n)^n＜3(n为正整数)怎么做,

Question

求教一道二项式定理的题目
求证:2＜＝(1+1/n)^n＜3(n为正整数)怎么做,

Answer 1

(1+1/n)^n
=1+n*(1/n)+[n(n-1)/2]*(1/n)²+...
=1+1+(1-1/n)2+.
=2+(1-1/n)2+.
≥2
令n->∞,
(1+1/n)^n --> e = 2.71828...≤3
∴ 2 ≤ (1+1/n)^n ≤ 3