数列求和:1/2+3/4+5/8+7/16+.+(2n-1)/(2^n)

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数列求和:1/2+3/4+5/8+7/16+.+(2n-1)/(2^n)

数列求和:1/2+3/4+5/8+7/16+.+(2n-1)/(2^n)
数列求和:1/2+3/4+5/8+7/16+.+(2n-1)/(2^n)

数列求和:1/2+3/4+5/8+7/16+.+(2n-1)/(2^n)
设原式=Sn
2*Sn=1+3/2+5/4+7/8+...+(2n-1)/(2^(n-1))
上式减去原式(错位相减)得出
Sn=1+2*1/2+2*1/4+2*1/8+...+2*1/(2^(n-1))-(2n-1)/(2^n)
=1-(2n-1)/(2^n)+[1+1/2+1/4+...+1/(2^(n-2))]
=1-(2n-1)/(2^n)+2-1/(2^(n-2))
=3-(2n-1+4)/(2^n)
=3-(2n+3)/(2^n)

-2*(1/2)^(n+1)-4*(1/2)^(n+1)*(n+1)+3