Power series from the geometric series
Problem 7.283 · medium
Find a power series for \( \displaystyle f(x) = \frac{x}{x^{2} + 1} \) centered at 0, and its interval of convergence.
- \[ \frac{x}{x^{2} + 1} \]Write f in the form (something)/(1 − u) with u = -x**2.✓ Proved
- 1/(1 − u) = Σ uⁿ for |u| < 1; substitute and multiply through.Reviewed
- \[ \lim_{x \to 0^+}\left(\frac{x^{7} - x^{5} + x^{3} - x + \frac{x}{x^{2} + 1}}{x^{7}}\right) = 0 \]The first terms -x**7 + x**5 - x**3 + x agree with f to order x^7.✓ Proved
- |u| < 1 means |x| < 1; the series diverges at both endpoints (the terms do not shrink).Reviewed
Answer \( \sum_{n=0}^{\infty} x \left(- x^{2}\right)^{n},\quad |x| < 1 \)
✓ Nihil obstat Lines: 2 proved, 2 reviewed. 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 |
| 2 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | the series summed numerically at half the radius matches f |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the geometric series structure, applies the substitution valid for |x| < 1, and correctly determines the interval of convergence by checking endpoint divergence.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the geometric series structure, applies the substitution valid for |x| < 1, and correctly determines the interval of convergence by checking endpoint divergence.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the geometric series structure, applies the substitution u = -x^2, and correctly determines the interval of convergence including the behavior at the endpoints.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/power_series_geometric, checked 2026-10-05 with SymPy 1.14.0.