Why does Σ uₙ diverge?

Σ 1/√n is a p-series with p = 1/2 ≤ 1, which diverges

Ratio test gives limit greater than 1

Σ 1/√n converges, so uₙ is bounded

Convergence of Σ n/(1+n√(n+1))

Short answer

Term behaves like 1/√n. Limit comparison with ∑1/√n gives limit 1; the p-series ∑1/√n diverges (p=1/2), so the series diverges.

Determine whether the series Σ n/(1+n√(n+1)) is convergent or divergent.

The series is Σ uₙ with uₙ = n/(1 + n√(n + 1)).

In the denominator, n√(n + 1) ≈ n^(3/2), so uₙ ≈ n/n^(3/2) = 1/√n.

uₙ/vₙ = n√n / (1 + n√(n+1))
      = 1 / (1/n^(3/2) + √(1 + 1/n))
      → 1 / (0 + 1) = 1

The limit is finite and positive, so Σ uₙ and Σ vₙ behave alike. Σ 1/√n = Σ 1/n^(1/2) is a p-series with p = 1/2 ≤ 1, which diverges.

Answer: the series Σ n/(1 + n√(n + 1)) diverges.

✓ Answer checked by Claude Opus 5.5 (high effort)