Verified answer

1/(n² - 1) = 1 / (n - 1)(n + 1) = 1/2 *( (n + 1) - (n - 1))/ (n - 1)(n + 1) = 1/2*(1 / (n - 1) - 1/(n + 1))

1/3 + 1/8 + 1/15 + ..... + 1/(n² - 1) = 1/1*3 + 1/2*4 + 1/3*5 + .... + 1/(n -1)(n +1) = 1/2(1 - 1/3 + 1/2 - 1/4 + .... + 1/(n - 2) - 1/n + 1/(n - 1) - 1/(n + 1)) = 1/2*(1 + 1/2 - 1/(n - 1) - 1/n) = 1/2 * (3/2 - (2n-1)/(n-1)n )