Error bounds for series
Problem 7.383 · easy
How many terms of \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{n!} \) 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}.
- \[ \left. \frac{1}{n!} \right|_{\substack{ n=7 }} = \frac{1}{5040} \]b_7 = 1/5040 < 1/1000.✓ Proved
- \[ \left. \frac{1}{n!} \right|_{\substack{ n=6 }} = \frac{1}{720} \]but b_6 = 1/720 ≥ 1/1000, so N = 6 is the smallest that works.✓ Proved
Answer \( N = 6 \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | 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: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies N=6 as the smallest N. Since b_6 = 1/720 > 1/1000, the error bound for N=5 is not guaranteed to be below 1/1000,
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies N=6 as the smallest N. Since b_6 = 1/720 > 1/1000, the error bound for N=5 is not guaranteed to be below 1/1000,gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution concludes N=6 is the smallest integer satisfying the condition, but fails to check N=5. Since b_6 = 1/720 < 1/1000, N=5 is sufficient and is the correct answer.
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 2026-10-06 with SymPy 1.14.0.