The sum of 1 / 1 × 2 + 1 / 2 × 3 +. + 1 / N (n + 1) is obtained by the split term elimination method Like the title,

1/(1×2)+1/(2×3)+...+1/[n(n+1)]
=1-1/2+1/2-1/3+...+1/n -1(n+1)
=1- 1/(n+1)
=n/(n+1)

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