∞

 ∑ 2/(n^2 - 1)

n=2

  2/(n^2 - 1)

  2/((n + 1)(n - 1))

  2/((n + 1)(n - 1)) = A/(n + 1) + B/(n - 1)

  2(n + 1)/((n + 1)(n - 1)) = A + B(n + 1)/(n - 1)

  2/(n - 1) = A + B(n + 1)/(n - 1)

  2/(-1 - 1) = A + B(-1 + 1)/(-1 - 1)

  2/(-2) = A + B(0)/(-2)

  [-1] = A

  2(n - 1)/((n + 1)(n - 1)) = A(n - 1)/(n + 1) + B

  2/(n + 1) = A(n - 1)/(n + 1) + B

  2/(1 + 1) = A(1 - 1)/(1 + 1) + B

  2/(2) = A(0)/(2) + B

  [1] = B

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

 N

 ∑ A/(n + 1) + B/(n - 1)

n=2

 N

 ∑ -1/(n + 1) + 1/(n - 1)

n=2

   ┌── n = 2 ─┐   ┌── n = 3 ─┐   ┌── n = 4 ─┐    ┌── n = 5 ─┐       ┌──────── n = N ──────┐

 = (-1/3 + 1/1) + (-1/4 + 1/2) + (-1/5 + 1/3)  + (-1/6 + 1/4) ... + (-1/(N + 1) + 1/(N - 1)

   ┌── n = 2 ─┐   ┌── n = 3 ─┐   ┌── n = 4 ─┐    ┌── n = 5 ─┐       ┌──────── n = N ──────┐

 = (-1/3 + 1/1) + (-1/4 + 1/2) + (-1/5 + 1/3)  + (-1/6 + 1/4) ... + (-1/(N + 1) + 1/(N - 1)

   ┌── n = 2 ─┐   ┌── n = 3 ─┐   ┌── n = 4 ─┐    ┌── n = 5 ─┐       ┌──────── n = N ──────┐

 = (-1/3 + 1/1) + (-1/4 + 1/2) + (-1/5 + 1/3)  + (-1/6 + 1/4) ... + (-1/(N + 1) + 1/(N - 1)

   ┌── n = 2 ─┐   ┌── n = 3 ─┐   ┌── n = 4 ─┐    ┌── n = 5 ─┐       ┌──────── n = N ───────┐   ┌────────────── n = (N + 1) ───────┐

 = (-1/3 + 1/1) + (-1/4 + 1/2) + (-1/5 + 1/3)  + (-1/6 + 1/4) ... + (-1/(N + 1) + 1/(N - 1)) + (-1/((N + 1) + 1) + 1/((N + 1) - 1))

   ┌── n = 2 ─┐   ┌── n = 3 ─┐   ┌── n = 4 ─┐    ┌── n = 5 ─┐       ┌──────── n = N ───────┐   ┌────────────── n = (N + 1) ───────┐

 = (-1/3 + 1/1) + (-1/4 + 1/2) + (-1/5 + 1/3)  + (-1/6 + 1/4) ... + (-1/(N + 1) + 1/(N - 1)) + (-1/((N + 1) + 1) + 1/((N + 1) - 1))

S = lim 1/1 + 1/2 - 1/(N + 1) - 1/((N + 1) + 1)

    N→∞

S = 1/1 + 1/2 - 1/(∞ + 1) - 1/((∞ + 1) + 1)

S = 1/1 + 1/2 - 1/(∞ + 1) - 1/(∞ + 1)

S = 1/1 + 1/2 - 1/∞ - 1/∞

S = 1/1 + 1/2 - 0 - 0

S = 1/1 + 1/2

S = 2/2 + 1/2

S = [3/2]