Error bounds for series
Problem 7.386 · easy
Use the integral test bound \( \displaystyle R_N \le \int_N^\infty f(x)\,dx \) to find the smallest \( \displaystyle N \) for which the partial sum \( \displaystyle S_N \) of \( \displaystyle \sum 1/n^{3} \) is within \( \displaystyle \frac{1}{10000} \) of the sum.
- \[ \int\limits_{71}^{\infty} \frac{1}{x^{3}}\, dx = \frac{1}{10082} \]With N = 71 the bound is 1/10082 < 1/10000.✓ Proved
- \[ \int\limits_{70}^{\infty} \frac{1}{x^{3}}\, dx = \frac{1}{9800} \]With N = 70 it is 1/9800, not small enough.✓ Proved
Answer \( N = 71 \)
Every line of this solution was proved by the computer algebra system SymPy. 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ 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: fail (error) — The solution checks the bound for N=71 and N=70 but fails to explicitly verify that N=71 is the *smallest* integer satisfying the condition. While the numerical check implies it, a rigorous solution must state that since the bound is a decreasing function of N, and N=70 fails while N=71 succeeds, N=71 is the minimum. More critically, the problem asks for the smallest N such that the partial sum is within 1/10000 of the sum. The integral test bound states R_N <= integral. If the integral is < 1/10000, then R_N < 1/10000. The logic holds, but the presentation is incomplete as it doesn't explicitly conclude 'therefore 71 is the smallest' based on the monotonicity of the bound. However, looking closer, is N=71 actually the smallest? Let's check N=70. Integral from 70 is 1/(2*70^2) = 1/9800 > 1/10000. So 70 fails. 71 works. The logic is sound, but the 'sentence' part is missing entirely. The prompt says 'Each line... is either an EQUATION... or a SENTENCE'. The provided solution has NO sentences. It just has two equations with comments. The comments are not formal sentences setting up the model or concluding. The review instructions say 'The sentences are what you are here for.' If there are no sentences, can I judge? The prompt implies I should check the sentences. If there are none, the solution is incomplete in its explanation. But usually, these problems expect a logical flow. Let's look at the 'Stated answer'. The solution provides the answer via calculation. Is it 'wrong'? No. Is it 'misleading'? No. Is it 'style'? Maybe. But wait, the prompt says 'Review it.' and 'Reply with JSON'. If the solution is just two equations, it lacks the necessary logical glue (sentences) to be a complete 'solution' in the context of 'checking sentences'. However, often in these datasets, the 'comments' attached to equations are treated as the reasoning. Let's assume the comments are the 'sentences' or the reasoning. The reasoning is: 'With N=71 bound is < 1/10000. With N=70 it is not.' This implies 71 is the smallest integer >= 70.5? No, it implies 71 is the first integer where the bound holds, assuming monotonicity. The monotonicity is not stated. This is a 'style' or 'misleading' issue if a student doesn't know why we don't check 69. But 69 would be worse. The bound is decreasing. So if 70 fails, all N<70 fail. This is a standard property. The omission of 'since the bound is decreasing' is a minor style/clarity issue, not an error. However, looking at the strict instructions: 'Check that... each conclusion follows from the lines before it.' The conclusion 'N=71' does not explicitly follow from the two equations without the implicit assumption of monotonicity and integer minimality. This is a gap in logic. I will mark it as 'style' or 'misleading' because it skips the justification for 'smallest'. Actually, let's look at the severity definitions. 'Error': step or setup is wrong. 'Misleading': student would learn something false. 'Style': correct but could be clearer. Skipping the monotonicity argument is a clarity issue. It's not false. It's not an error in calculation. It's a 'style' issue in rigor. But wait, is there a bigger error? The problem asks for the smallest N. The solution shows 71 works and 70 doesn't. This is sufficient to prove 71 is the smallest *if* one knows the bound is monotonic. Most calculus students know this. I will mark it as 'pass' if the comments are considered sufficient reasoning. But the prompt says 'The sentences are what you are here for.' and there are no explicit sentences. This is tricky. Let's look at the provided solution format again. It has comments. I will treat the comments as the reasoning. The reasoning is incomplete. It doesn't state why 71 is the *smallest*. It just shows 71 works and 70 doesn't. A student might think they need to check 69, 68, etc. The logical jump is missing. I'll call it 'style' for lack of explicit justification of minimality.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The solution checks the bound for N=71 and N=70 but fails to explicitly verify that N=71 is the *smallest* integer satisfying the condition. While the numerical check implies it, a rigorous solution must state that since the bound is a decreasing function of N, and N=70 fails while N=71 succeeds, N=71 is the minimum. More critically, the problem asks for the smallest N such that the partial sum is within 1/10000 of the sum. The integral test bound states R_N <= integral. If the integral is < 1/10000, then R_N < 1/10000. The logic holds, but the presentation is incomplete as it doesn't explicitly conclude 'therefore 71 is the smallest' based on the monotonicity of the bound. However, looking closer, is N=71 actually the smallest? Let's check N=70. Integral from 70 is 1/(2*70^2) = 1/9800 > 1/10000. So 70 fails. 71 works. The logic is sound, but the 'sentence' part is missing entirely. The prompt says 'Each line... is either an EQUATION... or a SENTENCE'. The provided solution has NO sentences. It just has two equations with comments. The comments are not formal sentences setting up the model or concluding. The review instructions say 'The sentences are what you are here for.' If there are no sentences, can I judge? The prompt implies I should check the sentences. If there are none, the solution is incomplete in its explanation. But usually, these problems expect a logical flow. Let's look at the 'Stated answer'. The solution provides the answer via calculation. Is it 'wrong'? No. Is it 'misleading'? No. Is it 'style'? Maybe. But wait, the prompt says 'Review it.' and 'Reply with JSON'. If the solution is just two equations, it lacks the necessary logical glue (sentences) to be a complete 'solution' in the context of 'checking sentences'. However, often in these datasets, the 'comments' attached to equations are treated as the reasoning. Let's assume the comments are the 'sentences' or the reasoning. The reasoning is: 'With N=71 bound is < 1/10000. With N=70 it is not.' This implies 71 is the smallest integer >= 70.5? No, it implies 71 is the first integer where the bound holds, assuming monotonicity. The monotonicity is not stated. This is a 'style' or 'misleading' issue if a student doesn't know why we don't check 69. But 69 would be worse. The bound is decreasing. So if 70 fails, all N<70 fail. This is a standard property. The omission of 'since the bound is decreasing' is a minor style/clarity issue, not an error. However, looking at the strict instructions: 'Check that... each conclusion follows from the lines before it.' The conclusion 'N=71' does not explicitly follow from the two equations without the implicit assumption of monotonicity and integer minimality. This is a gap in logic. I will mark it as 'style' or 'misleading' because it skips the justification for 'smallest'. Actually, let's look at the severity definitions. 'Error': step or setup is wrong. 'Misleading': student would learn something false. 'Style': correct but could be clearer. Skipping the monotonicity argument is a clarity issue. It's not false. It's not an error in calculation. It's a 'style' issue in rigor. But wait, is there a bigger error? The problem asks for the smallest N. The solution shows 71 works and 70 doesn't. This is sufficient to prove 71 is the smallest *if* one knows the bound is monotonic. Most calculus students know this. I will mark it as 'pass' if the comments are considered sufficient reasoning. But the prompt says 'The sentences are what you are here for.' and there are no explicit sentences. This is tricky. Let's look at the provided solution format again. It has comments. I will treat the comments as the reasoning. The reasoning is incomplete. It doesn't state why 71 is the *smallest*. It just shows 71 works and 70 doesn't. A student might think they need to check 69, 68, etc. The logical jump is missing. I'll call it 'style' for lack of explicit justification of minimality.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the integral test remainder bound formula and verifies the inequality for N=71 and N=70 to determine the smallest integer N.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.