Question

n＞1，n属于正整数，证明(1+1/3）(1+1/5)(1+1/7)…(1+1/(2n—1)＞√（2n+1）/2 用综合法 。

Answer 1

[(1+1/3）(1+1/5)(1+1/7)…(1+1/(2n-1)]²
=(1+1/3)(1+1/3)(1+1/5)(1+1/5)(1+1/7)(1+1/7).......(1+1/(2n-1))*(1+1/(2n-1))
>(1+1/3)(1+1/4)(1+1/5)(1+1/6)(1+1/7)(1+1/8).......(1+1/(2n-1))*(1+1/2n)
=(4/3)(5/4)(6/5)(7/6)*(8/7)*(9/8).........(2n/(2n-1)) *(2n+1)/2n
=(2n+1)/3
>(2n+1)/4
所以 (1+1/3）(1+1/5)(1+1/7)…(1+1/(2n—1)＞√（2n+1）/2

Answer 2

因为 4n²>4n²-1=(2n+1)(2n-1)
所以 2n>√[(2n+1)(2n-1)]
2n/(2n-1)>√(2n+1)/√(2n-1)
所以 (1+1/3）(1+1/5)(1+1/7)…[1+1/(2n—1)]
=(4/3)(6/5)...[2n/(2n-1)]
>(√5/√3)(√7/√5)...[√(2n+1)/√(2n-1)]
=√(2n+1)/√3
>√(2n+1)/2