已知数列an中an=2^n/(2^n-1) 求证∑ai(ai-1)

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已知数列an中an=2^n/(2^n-1) 求证∑ai(ai-1)

已知数列an中an=2^n/(2^n-1) 求证∑ai(ai-1)
已知数列an中an=2^n/(2^n-1) 求证∑ai(ai-1)<3

已知数列an中an=2^n/(2^n-1) 求证∑ai(ai-1)
因为an=2^n/(2^n-1) ,
从而ai(ai-1)=(2^i/(2^i-1))* (1/(2^i-1)) ;
(2^i-1))* (2^i-1)=2^(2i)-2^(i+1)+1
当i>=2时
(2^i-1))* (2^i-1)=2^(2i)-2^(i+1)+1>2^(2i-1)
ai(ai-1)<2^i/(2^(2i-1)=1/(2^(i-1)
∑ai(ai-1)<∑1/(2^(i-1)<3