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Convergence of x/1.2 + x²/2.3 + x³/3.4

Short answer

Converges for 0 < x ≤ 1, diverges for x > 1. Ratio test gives limit x; at x = 1, series telescopes to1 and converges.

Test the convergence of the series x/1.2 + x²/2.3 + x³/3.4 + ..., x > 0.

The series x/(1·2) + x²/(2·3) + x³/(3·4) + … has general term

uₙ = xⁿ / (n(n + 1)), n = 1, 2, 3, …

Ratio test

uₙ₊₁/uₙ = x · n(n+1) / ((n+1)(n+2)) = x · n/(n+2) → x
  • If x < 1, the series converges.
  • If x > 1, the series diverges.

The case x = 1

uₙ = 1/(n(n + 1)) = 1/n − 1/(n + 1). Compare with Σ 1/n²: limit of uₙ/(1/n²) = n²/(n(n+1)) → 1 (finite and non-zero), and Σ 1/n² converges, so the series converges (its sum is 1, by telescoping).

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

✓ Answer checked by Claude Opus 5.5 (high effort)

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