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Error bounds for series

Problem 7.300 · 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 \( \displaystyle \sum 1/n^{2} \) is within \( \displaystyle \frac{1}{1000} \) of the sum.

\[ \int\limits_{1001}^{\infty} \frac{1}{x^{2}}\, dx = \frac{1}{1001} \]
With N = 1001 the bound is 1/1001 < 1/1000.✓ Proved
\[ \int\limits_{1000}^{\infty} \frac{1}{x^{2}}\, dx = \frac{1}{1000} \]
With N = 1000 it is 1/1000, not small enough.✓ Proved

Answer \( N = 1001 \)

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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the integral test bound formula and identifies the smallest integer N satisfying the inequality.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly applies the integral test bound formula and identifies the smallest integer N satisfying the inequality.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to establish that N=1000 is insufficient. Since the bound for N=1000 is exactly 1/1000, it does not satisfy the strict inequality R_N < 1/1000 required to be 'within' 1/1000. The solution merely states it is 'not small enough' without proving that no smaller N works or that N=1000 is the immediate predecessor, leaving the 'smallest' claim unjustified.
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.