Use the Limit Comparison Test to show that ∞Σ(n=1) (n+1)/(n+2)^3 converges, and that ∞Σ(n=1) (n+1)/(n+2)^2 diverges.?

0 < (n + 1)/(n + 2)³ < (n + 1)/(n + 1)³ = 1/(n + 1)²
0 < Σ(n = 1,∞) 1/(n + 1)² < Σ(n = 1,∞) 1/(n + 1)²

∵ Σ(n = 1,∞) 1/(n + 1)² converges ...... by the integral test
∴ Σ(n = 1,∞) 1/(n + 1)² converges

(n + 1)/n² < (n + 1)/(n + 2)² < (n + 1)/(n + 1)² = 1/(n + 1)
Σ(n = 1,∞) (1/n + 1/n²) < Σ(n = 1,∞) (n + 1)/(n + 2)² < Σ(n = 1,∞) 1/(n + 1)

∵ Σ(n = 1,∞) (1/n + 1/n²) and Σ(n = 1,∞) 1/(n + 1) diverge ...... by the integral test
∴ Σ(n = 1,∞) (n + 1)/(n + 2)² diverges