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Error bounds for series

Problem 7.308 · 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 \( \displaystyle \frac{1}{100} \) of the sum.

\[ \int\limits_{8}^{\infty} \frac{1}{x^{3}}\, dx = \frac{1}{128} \]
With N = 8 the bound is 1/128 < 1/100.✓ Proved
\[ \int\limits_{7}^{\infty} \frac{1}{x^{3}}\, dx = \frac{1}{98} \]
With N = 7 it is 1/98, not small enough.✓ Proved

Answer \( N = 8 \)

✓ Nihil obstat 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. 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	✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the integral test remainder bound formula and verifies the inequality for N=8 and N=7, confirming N=8 is the smallest integer satisfying the condition.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the integral test remainder bound formula and verifies the inequality for N=8 and N=7, confirming N=8 is the smallest integer satisfying the condition.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the integral test remainder bound formula and verifies the inequality for N=8 and N=7 to identify the smallest integer satisfying the condition.
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.