Error bounds for series
Problem 7.299 · easy
How many terms of \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{n^{2}} \) guarantee an error below \( \displaystyle \frac{1}{1000} \)? Use the alternating series estimate \( \displaystyle |R_N| \le b_{N+1} \).
- The error after N terms is at most the first omitted term, b_{N+1}.Reviewed
- \[ \left. \frac{1}{n^{2}} \right|_{\substack{ n=32 }} = \frac{1}{1024} \]b_32 = 1/1024 < 1/1000.✓ Proved
- \[ \left. \frac{1}{n^{2}} \right|_{\substack{ n=31 }} = \frac{1}{961} \]but b_31 = 1/961 ≥ 1/1000, so N = 31 is the smallest that works.✓ Proved
Answer \( N = 31 \)
✓ Nihil obstat Lines: 2 proved, 1 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ 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 | the bound recomputed in floating point at N and N − 1 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly applies the Alternating Series Estimation Theorem and identifies the smallest N satisfying the error bound.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the Alternating Series Estimation Theorem and identifies the smallest N satisfying the error bound.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the Alternating Series Estimation Theorem. It identifies that N=31 is the smallest integer such that the first omitted term b_{32} is less than 1/1000, while b_{31} is not.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/series_error_bounds, checked 2026-10-05 with SymPy 1.14.0.