Convergence of 1+1²/2²+2²/3³+...

The series is 1 + 1²/2² + 2²/3³ + 3³/4⁴ + 4⁴/5⁵ + …

The pattern of the terms after the first: 1/4, 4/27, 27/256, 256/3125, so the general term is

uₙ = nⁿ/(n + 1)ⁿ⁺¹, n = 1, 2, 3, … (with 1 as the first term, n = 0)

Write uₙ in a convenient form

uₙ = (1/(n+1)) · (n/(n+1))ⁿ = (1/(n+1)) · (1 + 1/n)⁻ⁿ

Since (1 + 1/n)ⁿ → e, we have (1 + 1/n)⁻ⁿ → 1/e.

Limit comparison test with vₙ = 1/n

uₙ/vₙ = (n/(n+1)) (1 + 1/n)⁻ⁿ → 1 · (1/e) = 1/e

This limit is finite and non-zero. Σ 1/n is the harmonic series, which diverges, so Σ uₙ also diverges.

(The ratio test is inconclusive here: uₙ₊₁/uₙ → 1.)

Answer: the series diverges.