Answer:
Convergent
Step-by-step explanation:
∑₁°° n / (n² + 1)²
Applying integral test:
∫₁°° x / (x² + 1)² dx
If u = x² + 1, then du = 2x dx, or ½ du = x dx.
When x = 1, u = 2. When x = ∞, u = ∞.
½ ∫₂°° 1 / u² du
½ (-1 / u) |₂°°
½ (-1 / ∞) − ½ (-1 / 2)
0 + ¼
¼
The integral converges, so the series also converges.
