Home›Calculus 2›Error bounds for series›Problem 7.387

Error bounds for series

Problem 7.387 · 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 \( \displaystyle \frac{1}{100} \) of the sum.

\[ \int\limits_{101}^{\infty} \frac{1}{x^{2}}\, dx = \frac{1}{101} \]
With N = 101 the bound is 1/101 < 1/100.✓ Proved
\[ \int\limits_{100}^{\infty} \frac{1}{x^{2}}\, dx = \frac{1}{100} \]
With N = 100 it is 1/100, not small enough.✓ Proved

Answer \( N = 101 \)

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 R_N <= 1/N and identifies N=101 as the smallest integer satisfying the condition R_N < 1/100.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the integral test bound R_N <= 1/N and identifies N=101 as the smallest integer satisfying the condition R_N < 1/100.
qwen3.6:27b-mlx: inconclusive 2026-10-06 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution fails to establish that N=100 is insufficient by checking the strict inequality required for the bound to be strictly less than 1/100,
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.