Home›Calculus 2›Power series from the geometric series›Problem 7.281

Power series from the geometric series

Problem 7.281 · medium

Find a power series for \( \displaystyle f(x) = \frac{x}{x + 1} \) centered at 0, and its interval of convergence.

\[ \frac{x}{x + 1} \]
Write f in the form (something)/(1 − u) with u = -x.✓ Proved
1/(1 − u) = Σ uⁿ for |u| < 1; substitute and multiply through.
Reviewed
\[ \lim_{x \to 0^+}\left(\frac{x^{4} - x^{3} + x^{2} - x + \frac{x}{x + 1}}{x^{4}}\right) = 0 \]
The first terms -x**4 + x**3 - x**2 + x agree with f to order x^4.✓ Proved
|u| < 1 means |x| < 1; the series diverges at both endpoints (the terms do not shrink).
Reviewed

Answer \( \sum_{n=0}^{\infty} x \left(- x\right)^{n},\quad |x| < 1 \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the geometric series expansion and the interval of convergence. The reasoning for endpoint divergence is sound.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly identifies the geometric series expansion and the interval of convergence. The reasoning for endpoint divergence is sound.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly derives the power series and interval of convergence. The reasoning for endpoint divergence is sound.
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.