Convergence of series (n²−1)/(n²+1)xⁿ

[The terms printed are x, (3/5)x², (8/10)x³, (15/17)x⁴, …; coefficients 3/5, 8/10, 15/17 fit (n² − 1)/(n² + 1) for n = 2, 3, 4. The first term printed as x is read as the n = 1 term, which does not change convergence.]

General term: uₙ = (n² − 1)/(n² + 1) · xⁿ.

Ratio test

uₙ₊₁/uₙ = x · [((n+1)² − 1)/((n+1)² + 1)] · [(n² + 1)/(n² − 1)]

As n → ∞ each bracket tends to 1, so the ratio → x.

• x < 1: converges.
• x > 1: diverges.

The case x = 1

uₙ = (n² − 1)/(n² + 1) = 1 − 2/(n² + 1) → 1 ≠ 0. Since the general term does not tend to zero, the series diverges.

Answer: converges for 0 < x < 1; diverges for x ≥ 1.