作者gkaok2 (gkaok2)
看板trans_math
标题Re: [微分] 一题证明
时间Tue Feb 9 08:33:59 2010
※ 引述《BIEGABOY (BIEGABOY)》之铭言：
: Let the sequence an=(1+1/n)^n
: 1.show that an is increasing.
Proof:
Let f(x)=(1+1/x)^x f(x_n)=a_n x_1=1,x_2=2...,x_n=n
取自然对数 F(x)=ln[f(x)]=x*ln(1+1/x)=x*ln[(x+1)/x ] (里面变假分式)
递增递减性仍与原函数相同
x x-(x+1)
考虑 F'(x)=ln[(x+1)/x ]+x* -------------- * --------------
x + 1 x^2
=ln[(x+1)/x ] - (1/x+1)
claim : F'(x) > 0 for all x>=1
since 1. F'(1)=ln2-1/2 >0
2. F"(x)= (x/x+1) - (-1)*(x+1)^(-2) > 0
所以F'(x) >0 => F(x) 递增 => f(x)递增
=> f(x_n+1) > f(x_n)
=> a_n+1 > a_n for all n>=1
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※ 编辑: gkaok2 来自: 140.113.22.70 (02/09 08:53)