数学的数列求和 dn=2^(2n-1)+1,求证1/(d2-d1)+1/(d3-d2)+...+1/(d(n+1)-dn)

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数学的数列求和 dn=2^(2n-1)+1,求证1/(d2-d1)+1/(d3-d2)+...+1/(d(n+1)-dn)
数学的数列求和 dn=2^(2n-1)+1,求证1/(d2-d1)+1/(d3-d2)+...+1/(d(n+1)-dn)<1/3

数学的数列求和 dn=2^(2n-1)+1,求证1/(d2-d1)+1/(d3-d2)+...+1/(d(n+1)-dn)
由通项可得
1/(d2-d1)+1/(d3-d2)+...+1/(d(n+1)-dn)
=1/(2^3-2^1)+1/(2^5-2^3)+1/(2^7-2^5)+.+1/[2^(2n+1)-2^(2n-1)]
=1/3*[1/2+1/2^3+1/2^5+.+1/2^(2n-1)]
=1/3*1/2*[1-(1/2)^(2n)]/(1-1/4)
=2/9*[1-(1/2)^(2n)]
<2/9
<3/9
=1/3 .

由dn=2^(2n-1)+1得到dn+1=2^(2n+1)+1那么
(dn+1)-dn=2^(2n+1)+1-(2^(2n-1)+1)=3*2^(2n-1)
1/(d(n+1)-dn)=1/3*2^(2n-1)=1/3*1/2^(2n-1)
1/2^(2n-1)<1则1/3*1/2^(2n-1)<1/3则
1/(d2-d1)+1/(d3-d2)+...+1/(d(n+1)-dn)<1/3