求数列1/1,1/1+2,1/1+2+3,。。。1/1+2+3...+n的前n项和
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Sn=1/1+1/(1+2)+1/(1+2+3)+......+1/(1+2+3+....n)
=2/(1*2)+2/(2*3)+2/(3*4)+.....+2/[n(n+1)]
=2{1/(1*2)+1/(2*3)+1/(3*4)+....1/[n(n+1)]}
=2[1-1/2+1/2-1/3+1/3-1/4+....1/n-1/(n+1)]
=2[1-1/(n+1)]
=2n/(n+1)
=2/(1*2)+2/(2*3)+2/(3*4)+.....+2/[n(n+1)]
=2{1/(1*2)+1/(2*3)+1/(3*4)+....1/[n(n+1)]}
=2[1-1/2+1/2-1/3+1/3-1/4+....1/n-1/(n+1)]
=2[1-1/(n+1)]
=2n/(n+1)
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