Power series from the geometric series
Problem 7.284 · medium
Find a power series for \( \displaystyle f(x) = \frac{1}{2 x + 1} \) centered at 0, and its interval of convergence.
- \[ 1 \frac{1}{2 x + 1} = \frac{1}{2 x + 1} \]Write f in the form (something)/(1 − u) with u = -2*x.✓ Proved
- 1/(1 − u) = Σ uⁿ for |u| < 1; substitute and multiply through.Reviewed
- \[ \lim_{x \to 0^+}\left(\frac{8 x^{3} - 4 x^{2} + 2 x - 1 + \frac{1}{2 x + 1}}{x^{3}}\right) = 0 \]The first terms -8*x**3 + 4*x**2 - 2*x + 1 agree with f to order x^3.✓ Proved
- |u| < 1 means |x| < 1/2; the series diverges at both endpoints (the terms do not shrink).Reviewed
Answer \( \sum_{n=0}^{\infty} \left(- 2 x\right)^{n},\quad |x| < \frac{1}{2} \)
✓ 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 form, determines the interval of convergence, and correctly notes divergence at the endpoints. The algebraic verification in step 3 is consistent with the derived series.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the geometric series form, determines the interval of convergence, and correctly notes divergence at the endpoints. The algebraic verification in step 3 is consistent with the derived series.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the geometric series form, derives the correct power series and interval of convergence, and properly handles the endpoint analysis.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.