简算1+1/(1+2)+1/(1+2+3)+.+1/(1+2+3+.+2009)

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简算1+1/(1+2)+1/(1+2+3)+.+1/(1+2+3+.+2009)
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简算1+1/(1+2)+1/(1+2+3)+.+1/(1+2+3+.+2009)
简算1+1/(1+2)+1/(1+2+3)+.+1/(1+2+3+.+2009)

简算1+1/(1+2)+1/(1+2+3)+.+1/(1+2+3+.+2009)
1/(1+2+...+n) = 2/(n)(n+1) = 2[1/n - 1/(n+1)]
1+1/3+1/6+1/10+1/15+.+1/1+2+3+...+n
=2 (1/1-1/2+1/2-1/3+1/3-1/4 + ...+1/n-1/n+1)
=2 [1-1/(n+1)]
=2n/(n+1)