1/2!+2/3!+3/4!+……+n/(n+1)!

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1/2!+2/3!+3/4!+……+n/(n+1)!

1/2!+2/3!+3/4!+……+n/(n+1)!
1/2!+2/3!+3/4!+……+n/(n+1)!

1/2!+2/3!+3/4!+……+n/(n+1)!
n/(n+1)!=(n+1-1)/(n+1)!=(1/n!)-(1/(n+1)!)
原式=(1/1!-1/2!)+(1/2!-1/3!)+...+(1/n!-1/(n+1)!)
=1/1!-1/(n+1)!
=1-1/(n+1)!

设1/2! +2/3!+3/4!+……+n/(n+1)!=x
x+1/2!+1/3!+1/4!+........1/(n+1)!=2/2!+3/3!+4/4!+......(n+1)/(n+1)!
=1+1/2!+1/3!+1/4!+.......1/(n)!
x=1-1/(n+1)!

可以拆分~
1-1/2+2*(1/2-1/3)+3*(1/3-1/4)+......+n*(1/n-1/(n+1))
=1+1/2+1/3+....1/n-n/(n+1)
=(n+1)/2(n-1)!-n/(n+1)

n/(n+1)!=[(n+1)-1]/(n+1)!=1/n!-1/(n+1)!,所以结果是1-1/(n+1)!,