Home›Calculus 2›Error bounds for series›Problem 7.306

Error bounds for series

Problem 7.306 · 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 \( \displaystyle \sum 1/n^{4} \) is within \( \displaystyle \frac{1}{100} \) of the sum.

\[ \int\limits_{4}^{\infty} \frac{1}{x^{4}}\, dx = \frac{1}{192} \]
With N = 4 the bound is 1/192 < 1/100.✓ Proved
\[ \int\limits_{3}^{\infty} \frac{1}{x^{4}}\, dx = \frac{1}{81} \]
With N = 3 it is 1/81, not small enough.✓ Proved

Answer \( N = 4 \)

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: fail (error) — The solution fails to establish that N=4 is the *smallest* integer satisfying the condition. It checks N=3 and N=4 but omits the necessary check for N=2, where the bound 1/24 is also greater than 1/100. Without verifying that N=2 fails, the claim that N=4 is the smallest valid N is unsupported.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to establish that N=4 is the *smallest* integer satisfying the condition. It checks N=3 and N=4 but omits the necessary check for N=2, where the bound 1/24 is also greater than 1/100. Without verifying that N=2 fails, the claim that N=4 is the smallest valid N is unsupported.
gpt-oss:20b: pass 2026-10-05
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution fails to verify that N=1 and N=2 do not satisfy the condition, so it does not prove that N=4 is the smallest such integer. Additionally, the integral test bound requires f(x) to be decreasing, continuous, and positive, which is true for 1/x^4, but the logic for finding the 'smallest' N is incomplete.
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.