Absolute and conditional convergence
Problem 7.329 · easy
Does \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1} n^{2}}{n^{2} + 4} \) converge absolutely, converge conditionally, or diverge?
- \[ \lim_{n \to \infty}\left(\frac{n^{2}}{n^{2} + 4}\right) = 1 \]The terms do not go to 0 (divergence test): the series diverges.✓ Proved
Answer \( \text{diverges} \)
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | SymPy's Sum.is_convergent on the series and on its absolute values |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the Divergence Test (n-th term test). Since the limit of the absolute value of the terms is 1, the limit of the terms themselves does not exist (oscillating between 1 and -1), and thus is not 0. Therefore, the series diverges.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the Divergence Test (n-th term test). Since the limit of the absolute value of the terms is 1, the limit of the terms themselves does not exist (oscillating between 1 and -1), and thus is not 0. Therefore, the series diverges.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the Divergence Test. Since the limit of the absolute value of the terms is 1 (non-zero), the terms themselves do not approach 0, implying divergence.gpt-oss:20b: pass 2026-10-05
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/alternating_series, checked 2026-10-05 with SymPy 1.14.0.