Differential Calculus

Consider the series:
1/ (3²-1) + 1/ (5²-1) + 1/ (7²-1) + . + 1/ (201)²-1)

The general term T? can be written for the r-th term starting with r=1 for 3, r=2 for 5.
T? = 1/ (2r+1)² - 1) = 1/ (2r+1-1) (2r+1+1) = 1/ (2r * (2r+2) = 1/4 * 1/ (r (r+1)
T? = 1/4 * (1/r - 1/ (r+1)

The sum of the first n terms is:
S? = Σ T? = 1/4 * Σ (1/r - 1/ (r+1) from r=1 to n
S? = 1/4 * [ (1 - 1/2) + (1/2 - 1/3) + . + (1/n - 1/ (n+1) ]
S? = 1/4 * (1 - 1/ (n+1)

The last term is (201)²-1, so 2r+1 = 201, which gives r = 100. So, n=100.
S? = 1/4 * (1 - 1/101) = 1/4 * (100/101) = 25/101.