Problem Analysis #5

11.21s
3,112 toks
Problem Statement

Let ([0,2],B([0,2]),λ)( [0,2], \mathcal{B}([0,2]), \lambda ) be the Lebesgue measure space.
For each n∈Nn \in \mathbb{N} define

fn(x):=∑k=1n2123456(x2) k−1(1−x2)3,x∈[0,2].f_n(x) := \sum_{k=1}^{n^2} 123456 \left(\frac{x}{2}\right)^{\,k-1} \left( 1 - \frac{x}{2} \right)^{3}, \qquad x \in [0,2].

Compute the value

I:=lim⁡n→∞∫02fn(x) dx.I := \lim_{n\to\infty} \int_0^2 f_n(x)\,dx.
Ground Truth
82304
Execution Trace
Reasoning Process

[THOUGHT] Rewrite the finite sum as a geometric series, simplify the integrand, change variables t=x/2 to integrate on [0,1], express the remaining term as a Beta integral, and take the limit as n→∞. Verify the closed form and the limit using Python. [/THOUGHT] [PYTHON] import sympy as sp

n = sp.symbols('n', positive=True, integer=True) m = n**2 In = 1234562(sp.Rational(1,3) - sp.Rational(2,1)/((m+1)(m+2)(m+3))) print("Closed form I_n:", sp.simplify(In)) print("Limit:", sp.limit(In, n, sp.oo))

Numerical sanity check

import mpmath as mp def In_numeric(N): f = lambda x: sum([123456*(x/2)**(k-1)*(1-x/2)3 for k in range(1,N2+1)]) return mp.quad(f, [0,2]) for N in [1,2,3,5,10]: print(N, In_numeric(N)) print("Target", 82304) [/PYTHON]

Tool Output
[PYTHON OUTPUT] Closed form I_n: 82304*n**2*(n**4 + 6*n**2 + 11)/(n**6 + 6*n**4 + 11*n**2 + 6) Limit: 82304 1 61728.0 2 79952.4571428571 3 81929.8909090909 5 82278.8766788767 10 82303.534613884 Target 82304 [/PYTHON OUTPUT]
Reasoning Process

[THOUGHT] The Python-verified closed form for the integral tends to 82304 as (n\to\infty), so (I=82304). [\boxed{82304}]

Final Model Output
82304

Comparative Analysis (P5) (Average across runs)

Avg Time Taken

Solar-Open-100B
550.5
Gemini-3-Pro-Preview
42.3
Kanana-2-30B-Thinking
38.5
naver-hyperclovax/HCX-007
34.1
GLM-4.5-Air
31.7
K-EXAONE-236B-A23B
29.5
Kanana-2-30B-Thinking-2601
21.1
Solar-Pro 2
19.7
model_d_r1
18.6
GPT-5.2 (high)
13.1
axk1
7.6
EXAONE-4.0-32B
6.5

Avg Token Usage

Solar-Open-100B
17025.7
Kanana-2-30B-Thinking
9881.7
Kanana-2-30B-Thinking-2601
6458.3
GLM-4.5-Air
5912.0
Gemini-3-Pro-Preview
5738.7
Solar-Pro 2
5609.7
naver-hyperclovax/HCX-007
4853.7
K-EXAONE-236B-A23B
4492.3
EXAONE-4.0-32B
4188.3
model_d_r1
3289.7
GPT-5.2 (high)
3264.0
axk1
2930.0