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Convergence of 1+x/2+2!x²/3²+...

Short answer

Converges for x < e, diverges for x ≥ e. Ratio test limit is x/e; at x = e, terms do not approach zero.

Test convergent or divergent of the series 1 + x/2 + (2!/3²)x² + (3!/4³)x³ + ... ∞

The series 1 + x/2 + (2!/3²)x² + (3!/4³)x³ + … has general term, starting from n = 0 with u₀ = 1,

uₙ = n! xⁿ / (n + 1)ⁿ.

Ratio test

uₙ₊₁/uₙ = [(n+1)! xⁿ⁺¹/(n+2)ⁿ⁺¹] · [(n+1)ⁿ/(n! xⁿ)]
        = x (n+1)ⁿ⁺¹/(n+2)ⁿ⁺¹
        = x [(n+1)/(n+2)]ⁿ⁺¹
        = x / (1 + 1/(n+1))ⁿ⁺¹
        → x/e

(because (1 + 1/m)ᵐ → e.)

  • x < e: converges.
  • x > e: diverges.

The case x = e

uₙ₊₁/uₙ = e / (1 + 1/(n+1))ⁿ⁺¹. Since (1 + 1/m)ᵐ < e for every m, this ratio is greater than 1, so the terms increase and cannot tend to 0. The series diverges.

Answer: the series converges for x < e (x > 0 assumed) and diverges for x ≥ e.

✓ Answer checked by Claude Opus 5.5 (high effort)

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