Which test suits uₙ = ((n+1)/(n+2))ⁿ xⁿ, and why?

Cauchy's root test, since uₙ is an nth power

At x = 1, uₙ = ((n+1)/(n+2))ⁿ. Why does the series diverge?

uₙ → e⁻¹ ≠ 0, so the terms do not tend to zero

Short answer

Root test: (u_n)^{1/n} → x; converges for 0 < x < 1. At x=1, nth term → e^{-1} ≠0, so diverges. Converges only for 0 < x <1.

Test the convergence of the series 1/2 + (2/3)x + (3/4)²x² + (4/5)³x³ + ...

The series 1/2 + (2/3)x + (3/4)²x² + (4/5)³x³ + … has general term, for n = 1, 2, 3, …,

uₙ = ((n + 1)/(n + 2))ⁿ xⁿ,

after the first term 1/2. Leaving out one term does not change convergence.

(uₙ)^(1/n) = ((n+1)/(n+2)) · x = (1 − 1/(n+2)) x → x

uₙ = (1 − 1/(n + 2))ⁿ = ((n+1)/(n+2))ⁿ → e⁻¹ ≠ 0 as n → ∞. Since uₙ does not tend to zero, the series diverges.

Answer: the series converges for 0 < x < 1 and diverges for x ≥ 1.

✓ Answer checked by Claude Opus 5.5 (high effort)