# Start the question
text Chapter 8.6 Question 25
text Evaluate the following sum
sum from n = 2 to infinity of 1 over (n^2 - 1)
text Answer: use partial fractions
n^2 - 1 = (n-1)(n+1)
1 over (n^2 - 1) = A over (n-1) + B over (n + 1)
= (A(n+1) + B (n-1)) over ((n-1)(n+1))
1 = A n + A + B n - B
text Equate coefficients
0 = A - B
1 = A + B
text add equations
1 = 2A
A = 1 over 2
B = - 1 over 2
1 over (n squared - 1) = 1 over (2(n-1)) - 1 over (2(n+1))
S _ N = sum from n = 2 to N of 1 over (n^2 - 1)
=
(1 over 2 - 1 over 6) + (1 over 4 - 1 over 8) + (1 over 6 - 1 over 10)
+ (1 over 8 - 1 over 12)
+ ... +
+ (1 over (2(N-3)) - 1 over (2(N-1)) )
+ (1 over (2(N-2)) - 1 over (2N) )
+ (1 over (2(N-1)) - 1 over (2(N+1)) )
=
1 over 2 + 1 over 4 - 1 over (2N) - 1 over (2(N+1))
limit as N to infinity of S_N = 3 over 4