With partial fractions:

1/(n + n²) = 1/(n(n+1)) = A/n + B/(n+1)

A(n+1) + Bn = 1

n = 0 gives A = 1, n = -1 gives B = -1

1/(n+n²) = 1/n - 1/(n+1)

Σ (n = 1 to ∞) 1/(n+n²) = Σ (n = 1 to ∞) 1/n - Σ (n = 1 to ∞) 1/(n+1)

= Σ (n = 1 to ∞) 1/n - Σ (n = 2 to ∞) 1/n

= 1/1 + Σ (n = 2 to ∞) 1/n - Σ (n = 2 to ∞) 1/n

= 1

Guessing this is the standard solution