1/(2*3)+1/(3*4)+1/(4*5)+1/(5*6)+...+1/(9…
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The series is telescoping.
∞ . . . . . . . . . . . . . ∞
Σ 1/((n+1)(n+2)) = Σ [1/(n+1) - 1/(n+2)]
n=1 . . . . . . . . . . . n=1
The Nth partial sum is
s_N = (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/(N+1) - 1/(N + 2)) = 1/2 - 1/(N+2).
Letting N -> ∞, the sum is s = 1/2.
∞ . . . . . . . . . . . . . ∞
Σ 1/((n+1)(n+2)) = Σ [1/(n+1) - 1/(n+2)]
n=1 . . . . . . . . . . . n=1
The Nth partial sum is
s_N = (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/(N+1) - 1/(N + 2)) = 1/2 - 1/(N+2).
Letting N -> ∞, the sum is s = 1/2.