[THOUGHT] We need to compute the limit: [ \lim_{n\to\infty} \int_{[0,n]} \left(1+\frac{205}{69n}x\right)^{-69n} e^{(-\frac{6}{7}x)},dx = \frac{p}{q}. ] The integral is over x from 0 to n. The integrand is: [ f_n(x) = \left(1+\frac{205}{69n}x\right)^{-69n} e^{-\frac{6}{7}x}. ] We recognize that as (n \to \infty), (\left(1+\frac{a}{n}x\right)^{-n} \to e^{-a x}) for a constant a. Here we have exponent (-69n) and factor (\frac{205}{69n}) inside. Let's rewrite: [ \left(1+\frac{205}{69n}x\right)^{-69n} = \left[\left(1+\frac{205}{69n}x\right)^{-69}\right]^n. ] But more directly, note the known limit: (\lim_{n\to\infty} \left(1+\frac{a}{n}\right)^{n} = e^{a}). However here we have ((1+\frac{b}{n}x)^{-n}) with b = 205/69? Wait exponent is -69n, base is (1+\frac{205}{69n}x). So: [ \left(1+\frac{205}{69n}x\right)^{-69n} = \exp\left[-69n \ln\left(1+\frac{205}{69n}x\right)\right]. ] For large n, (\ln(1+y) \approx y - y^2/2 + O(y^3)), with (y = \frac{205}{69n}x). Then: [ -69n \ln\left(1+\frac{205}{69n}x\right) = -69n \left(\frac{205}{69n}x - \frac{1}{2}\left(\frac{205}{69n}x\right)^2 + \dots\right) = -205x + \frac{205^2 x^2}{2\cdot 69 n} + O(1/n^2). ] Thus as (n\to\infty), the term tends to (e^{-205x}). So the integrand pointwise converges to: [ e^{-205x} e^{-\frac{6}{7}x} = e^{-(205 + \frac{6}{7})x} = e^{-\left(\frac{205\cdot7 + 6}{7}\right)x} = e^{-\frac{1435+6}{7}x} = e^{-\frac{1441}{7}x}. ] But wait: 205*7 = 1435, plus 6 = 1441, yes.

Thus pointwise limit: (f_n(x) \to e^{-\frac{1441}{7}x}) for each fixed x as (n\to\infty). However the integration domain is ([0,n]) which expands to infinity. So we need to compute: [ \lim_{n\to\infty} \int_0^n f_n(x),dx. ] We suspect by dominated convergence theorem (if we can find a dominating integrable function) that the limit equals (\int_0^\infty e^{-\frac{1441}{7}x},dx). But we must check if the integrand is dominated. For large n, (f_n(x)) is decreasing in n? Possibly monotone? Let's examine: For fixed x, ((1+\frac{205}{69n}x)^{-69n}) increases to (e^{-205x}) as n increases? Actually ((1+\frac{a}{n})^n) increases to e^a for a>0, so ((1+\frac{a}{n})^{-n}) decreases to e^{-a}. Here exponent is -69n, base >1 for x>0. Let's set a = 205/69 * x. Then ((1+\frac{a}{n})^{-n}) decreases to e^{-a} as n increases. But here we have exponent -69n, so factor ((1+\frac{205}{69n}x)^{-69n} = [(1+\frac{a}{n})^{-n}]^{69}). Since ((1+\frac{a}{n})^{-n}) decreases, raising to power 69 preserves monotonicity (since it's positive). So (f_n(x)) is decreasing in n for each x. Thus (f_n(x) \le f_1(x)) for n>=1? Actually decreasing means (f_{n+1}(x) \le f_n(x)). So (f_n(x) \le f_1(x)) for n>=1. So we can dominate by (f_1(x)) on ([0,\infty))? But domain is only up to n, but we can extend integrand to 0 beyond n. For dominated convergence on ([0,\infty)), we need integrable dominating function. Let's check if (f_1(x)) is integrable on ([0,\infty)). For n=1: [ f_1(x) = \left(1+\frac{205}{69}x\right)^{-69} e^{-\frac{6}{7}x}. ] As x→∞, ((1+\frac{205}{69}x)^{-69} \sim (\frac{205}{69}x)^{-69} = O(x^{-69})), multiplied by (e^{-\frac{6}{7}x}) which decays exponentially, so overall decays exponentially. So integrable. Thus dominated convergence applies (with dominating function (f_1(x)) for x in [0,∞) by setting f_n(x)=0 for x>n). Then limit equals integral of pointwise limit: [ \int_0^\infty e^{-\frac{1441}{7}x},dx = \left[-\frac{7}{1441} e^{-\frac{1441}{7}x}\right]_0^\infty = \frac{7}{1441}. ] Thus p/q = 7/1441. Check gcd(7,1441): 1441 = 7205 + 6? Actually 7205=1435, remainder 6. So gcd(7,1441)=? 1441 mod 7 = 6, 7 mod 6 = 1, 6 mod 1=0, so gcd=1. So p=7, q=1441 relatively prime. Then p^2 q = 7^2 * 1441 = 49 * 1441.

Compute 491441: 144150 = 72050, subtract 1441 = 70609. So p^2 q = 70609.

But we should verify with Python numeric simulation for large n to ensure limit is indeed 7/1441 ≈ 0.004857... Let's compute integral for large n numerically.

We'll write Python code to compute the integral for large n (say n=1000) using numerical integration (quad from scipy). Then compute limit as n increases.

We'll also compute 7/1441 to compare.

Let's do it. [/THOUGHT] [PYTHON] import numpy as np from scipy.integrate import quad

def integrand(x, n): return (1 + (205/(69*n))x)**(-69n) * np.exp(-6/7 * x)

def compute_integral(n): result, error = quad(integrand, 0, n, args=(n,)) return result

Compute for increasing n

ns = [10, 50, 100, 200, 500, 1000] for n in ns: val = compute_integral(n) print(f"n={n}: integral = {val:.10f}")

Expected limit

expected = 7/1441 print(f"Expected limit = {expected:.10f}") print(f"Difference for n=1000: {compute_integral(1000) - expected:.10e}") [/PYTHON]

[THOUGHT] The Python output shows unexpected behavior: for n=100 and larger, the integral evaluates to nearly zero (or exactly zero due to underflow?). That suggests numerical issues: the integrand becomes extremely small because ((1+\frac{205}{69n}x)^{-69n}) decays super-exponentially for large n? Wait, but for fixed x, as n increases, that factor tends to (e^{-205x}), which is not zero. However, for large n, the base (1+\frac{205}{69n}x) is close to 1, raising to power (-69n) yields extremely small values due to finite precision? Possibly underflow.

Let's check: For n=100, x=100 (upper limit), inside: (1+\frac{205}{69*100}*100 = 1+\frac{205}{69} \approx 1+2.971 = 3.971). Then ((3.971)^{-6900}) is astronomically small (~10^(-3300)), beyond double precision. So indeed numerical integration fails for large n.

Thus we cannot rely on direct numeric integration for large n. But our analytic argument using dominated convergence seems solid. Let's verify with moderate n where underflow isn't severe, like n=10 gave 0.0048647, n=50 gave 0.0048591, while expected limit is 0.0048577. The values are approaching the limit from above (since f_n decreases). So indeed limit appears to be 7/1441 ≈ 0.0048577377.

Thus p=7, q=1441, p^2 q = 49 * 1441 = 70609.

Let's compute 49*1441 exactly using Python to confirm. [/THOUGHT] [PYTHON] print(49 * 1441) [/PYTHON]

[THOUGHT] The product 49 * 1441 equals 70609, confirming our earlier calculation. Therefore, the final answer is 70609. [/THOUGHT] \boxed{70609}