Power series from the geometric series
Problem 7.371 · medium
Find a power series for \( \displaystyle f(x) = \frac{4}{4 - x} \) centered at 0, and its interval of convergence.
- \[ \frac{4}{4 - x} = \frac{1}{1 - \frac{x}{4}} \]Write f in the form (something)/(1 − u) with u = x/4.✓ Proved
- 1/(1 − u) = Σ uⁿ for |u| < 1; substitute and multiply through.Reviewed
- \[ \lim_{x \to 0^+}\left(\frac{- \frac{x^{3}}{64} - \frac{x^{2}}{16} - \frac{x}{4} - 1 + \frac{4}{4 - x}}{x^{3}}\right) = 0 \]The first terms x**3/64 + x**2/16 + x/4 + 1 agree with f to order x^3.✓ Proved
- |u| < 1 means |x| < 4; the series diverges at both endpoints (the terms do not shrink).Reviewed
Answer \( \sum_{n=0}^{\infty} \left(\frac{x}{4}\right)^{n},\quad |x| < 4 \)
✓ 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, applies the convergence condition for the ratio, and correctly determines the interval of convergence by checking the endpoints.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the geometric series form, applies the convergence condition for the ratio, and correctly determines the interval of convergence by checking the endpoints.gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the geometric series form, applies the convergence condition, and correctly determines the interval of convergence by checking endpoints.
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 2026-10-06 with SymPy 1.14.0.