Suppose the following limit is given:
where and are positive integers which are relatively prime. Find .
oindent
[THOUGHT] The limit is (\lim_{n\to\infty} \int_0^n \left(1 + \frac{205}{69n} x \right)^{-69n} e^{-\frac{6}{7}x} , dx).
Pointwise, as (n \to \infty), (\left(1 + \frac{205}{69n} x \right)^{-69n} \to e^{-205 x}).
Thus, the integrand converges pointwise to (e^{-205 x} e^{-\frac{6}{7} x} = e^{-\left(205 + \frac{6}{7}\right) x}).
The limit of the integral is (\int_0^\infty e^{-\left(205 + \frac{6}{7}\right) x} , dx = \frac{1}{205 + \frac{6}{7}} = \frac{1}{\frac{1435 + 6}{7}} = \frac{7}{1441}).
Verify (\gcd(7, 1441) = 1) and compute (p^2 q = 7^2 \times 1441 = 49 \times 1441). [/THOUGHT] [PYTHON] import math print(math.gcd(7, 1441)) print(49 * 1441) [/PYTHON]
[THOUGHT] The calculation confirms (\gcd(7,1441)=1) and (7^2 \times 1441 = 70609). The limit is (\frac{7}{1441}), so (p=7), (q=1441), and (p^2 q = 70609). [/THOUGHT] \boxed{70609}
Explanation
The reasoning begins by analyzing the pointwise limit of the integrand as (n \to \infty). For fixed (x), the term (\left(1 + \frac{205}{69n} x \right)^{-69n}) converges to (e^{-205 x}), since (\left(1 + \frac{a}{n}\right)^{n b} \to e^{a b}) with (a = \frac{205}{69} x) and (b = -69), yielding (a b = -205 x).
Thus, the integrand (\left(1 + \frac{205}{69n} x \right)^{-69n} e^{-\frac{6}{7} x}) converges pointwise to (e^{-205 x} e^{-\frac{6}{7} x} = e^{-\left(205 + \frac{6}{7}\right) x}).
The upper limit of integration is (n), which approaches infinity, so the limit of the integral equals (\int_0^\infty e^{-\left(205 + \frac{6}{7}\right) x} , dx).
Compute the exponent: (205 + \frac{6}{7} = \frac{205 \cdot 7 + 6}{7} = \frac{1435 + 6}{7} = \frac{1441}{7}).
The integral evaluates to (\int_0^\infty e^{-\frac{1441}{7} x} , dx = \frac{7}{1441}).
Here, (\frac{7}{1441}) is in lowest terms, as verified by (\gcd(7, 1441) = 1).
Thus, (p = 7) and (q = 1441), so (p^2 q = 7^2 \cdot 1441 = 49 \cdot 1441 = 70609).
Justification for interchanging limit and integral relies on the integrand being non-negative and the pointwise convergence to an integrable function, allowing application of the monotone convergence theorem.