Error bounds for series
Problem 7.392 · easy
How many terms of \( \displaystyle \sum_{n=1}^{\infty} \frac{\left(-1\right)^{n + 1}}{n^{3}} \) guarantee an error below \( \displaystyle \frac{1}{100} \)? 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}.
- \[ \left. \frac{1}{n^{3}} \right|_{\substack{ n=5 }} = \frac{1}{125} \]b_5 = 1/125 < 1/100.✓ Proved
- \[ \left. \frac{1}{n^{3}} \right|_{\substack{ n=4 }} = \frac{1}{64} \]but b_4 = 1/64 ≥ 1/100, so N = 4 is the smallest that works.✓ Proved
Answer \( N = 4 \)
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 claims N=4 is the smallest integer satisfying the condition, but the check for N=3 (b_4 = 1/64 >= 1/100) only shows N=3 fails. It fails
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 claims N=4 is the smallest integer satisfying the condition, but the check for N=3 (b_4 = 1/64 >= 1/100) only shows N=3 fails. It failsqwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution concludes N=4 is the smallest N that works, but the check for N=3 (b_4 = 1/64 >= 1/100) only shows N=3 fails. It fails to check if N=4 actually works by verifying b_5 < 1/100, which it does (1/125 < 1/100), but the logic presented is disjointed and implies N=4 is the answer without explicitly confirming the condition for N=4 holds, while the phrasing 'so N=4 is the smallest that works' relies on the implicit assumption that N=3 failed and N=4 succeeds, but the text structure is confusing. More critically, the prompt asks for the number of terms. If N=4 terms are used, the error is bounded by b_5. The solution calculates b_5 and b_4. It states b_4 >= 1/100, so N=3 (which would have error bound b_4) doesn't work. It states b_5 < 1/100, so N=4 (which has error bound b_5) works. The conclusion is correct, but the sentence 'so N=4 is the smallest that works' is derived from the previous line which discussed b_4. The logical flow is: Check N=3 -> bound is b_4 -> 1/64 >= 1/100 -> Fail. Check N=4 -> bound is b_5 -> 1/125 < 1/100 -> Pass. The solution presents the calculations but the sentence linking them is weak. However, looking closer, is there a factual error? No. Is it misleading? The sentence 'but b_4 = 1/64 >= 1/100, so N = 4 is the smallest that works' is logically incomplete because it doesn't explicitly state that N=4 works, only that N=3 doesn't. A student might think N=4 is the answer just because N=3 failed, without verifying N=4. But wait, line 2 already established b_5 < 1/100. So the information is there. Is it an error? No. Is it misleading? Yes, the causal link 'so' is weak. But let's look for a harder error. The problem asks for N such that |R_N| <= 1/100. |R_N| <= b_{N+1}. We need b_{N+1} < 1/100. For N=3, b_4 = 1/64 > 1/100. For N=4, b_5 = 1/125 < 1/100. So N=4 is correct. The solution is technically correct in its facts. However, the sentence 'so N=4 is the smallest that works' follows immediately from the statement about b_4. It skips the explicit confirmation that N=4 works, relying on the reader to connect line 2 and 3. This is a style/clarity issue, but arguably 'misleading' if a student thinks the failure of N=3 automatically makes N=4 the answer without checking N=4. Let's call it misleading.gpt-oss:20b: pass 2026-10-06
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.