已知数列8*1/1^2*3^2,8*2/3^2*5^2,……8*n/(2n-1)^2(2n+1)^2,若sn为该数列的前

8n/((2n-1)²(2n+1)²)
=((2n+1)²-(2n-1)²)/((2n-1)²(2n+1)²)
=1/(2n-1)²-1/(2n+1)²
所以
Sn= (8×1)/(1²×3²)+(8×2)/(3²×5²)+(8×3)/(5²×7²)...+ 8n/((2n-1)²(2n+1)²)
=1/1²-1/3²+1/3²-1/5²+1/5²-1/7²+...+1/(2n-1)²-1/(2n+1)²
=1-1/(2n+1)²
=(4n(n+1))/(2n+1)²