lim[1/2!+2/3!+…+n/(n+1)!] n→∞

n/(n+1)!=(n+1-1)/(n+1)!=1/n!-1/(n+1)!
lim[1/2!+2/3!+…+n/(n+1)!]
=lim[(1-1/2!)+(1/2!-1/3!)+...+(1/n!-1/(n+1)!)]
=lim[1-1/(n+1)!]
=1