Create an explicit formula for the partial sum? Does the series diverge or converge?

Question

Answer 1

As the denominator is of degree 2 > 1, we know that it converges.

Writing 2/(n^2 - 1) as 1/(n-1) - 1/(n+1), the sum becomes obvious.

All of the -1/(n+1) terms are eliminated by 1/(n-1) 2 terms later.

Thus, we are left with 1/(2-1) + 1/(3-1) = 1 + 1/2 = 3/2

If you wish to show a partial sum, we write

∑_k=2ⁿ 2/(n²-1) = ∑_k=2ⁿ 1/(n-1) - 1/(n+1) = ∑_k=1^n-1 1/k - 1/(k+2) = 1/1 + 1/2 + ∑_k=3^n-1 (1/k - 1/k) - 1/n - 1/n+1 = 3/2 - 1/n - 1/n+1

The limit of the partial sums is 3/2