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1/2!+2/3!+3/4!+4/5!+……+n/(n+1)!
= (2-1)/2!+(3-1)/3!+(4-1)/4!+(5-1)/5!+……+(n+1-1)/(n+1)!
= (2/2!-1/2!)+(3/3!-1/3!)+(4/4!-1/4!)+(5/5!-1/5!)+……+{(n+1)/(n+1)!-1/(n+1)!}
= (1-1/2!)+(1/2!-1/3!)+(1/3!-1/4!)+(1/4!-1/5!)+……+{1/n!-1/(n+1)!}
= 1 - 1/(n+1)!
= (2-1)/2!+(3-1)/3!+(4-1)/4!+(5-1)/5!+……+(n+1-1)/(n+1)!
= (2/2!-1/2!)+(3/3!-1/3!)+(4/4!-1/4!)+(5/5!-1/5!)+……+{(n+1)/(n+1)!-1/(n+1)!}
= (1-1/2!)+(1/2!-1/3!)+(1/3!-1/4!)+(1/4!-1/5!)+……+{1/n!-1/(n+1)!}
= 1 - 1/(n+1)!
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