数列求和,1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+……+1/[1+2+3+……+(n+1)]

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数列求和,1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+……+1/[1+2+3+……+(n+1)]

数列求和,1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+……+1/[1+2+3+……+(n+1)]
数列求和,
1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+……+1/[1+2+3+……+(n+1)]

数列求和,1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+……+1/[1+2+3+……+(n+1)]
最后一项分母是:[1+2+...+(n+1)] = [(n+1)*(n+2)]/2 ;则 1/[1+2+3+……+(n+1)] = 2/[(n+1)*(n+2)] = 2*[1/(n+1) - 1/(n+2)];所以原式:1/(1+2)+1/(1+2+3)+1/(1+2+3+4)+……+1/[1+2+3+……+(n+1)] = 2*{(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+...+[1/(n+1)-1/(n+2)]} = 2*[1/2-1/(n+2)] = 1-2/(n+2) = n/(n+2)