1/3+1/8+1/15+1/24...

1/3 + 1/8 + 1/15 + 1/24 + ... + 1/(n(n+2))= 1/(1x3) + 1/(2x4) + 1/(3x5) + 1/(4x6) + ... + 1/(n(n+2))注意 1/(n(n+2)) = (1/2) * 2 / [n(n+2)]= (1/2) * (n+2 - n) / [n(n+2)]= (1/2) * { (n+2)/[n(n+2)] - n/[n(n+2)] }= (1/2) * [1/n - 1/(n+2)]於是1/(1x3) + 1/(2x4) + 1/(3x5) + 1/(4x6) + ... + 1/(n(n+2))= ...(1/2) (1/1 - 1/3)
+ (1/2) (1/2 - 1/4)
+ (1/2) (1/3 - 1/5)
+ (1/2) (1/4 - 1/6)
+ .............................
+ (1/2) [1/(n-2) - 1/n]
+ (1/2) [1/(n-1) - 1/(n+1)]
+ (1/2) [1/n - 1/(n+2)] = (1/2) [1/1 + 1/2 - 1/(n+1) - 1/(n+2)]= (1/2) (3/2 - (2n+3)/[(n+1)(n+2)])= (1/2) [3(n+1)(n+2) - 2(2n+3)] / [2(n+1)(n+2)]= (3n² + 9n + 6 - 4n - 6) / [4(n+1)(n+2)]= n(3n + 5) / [4(n+1)(n+2)]