Problem Analysis #4

[THOUGHT] To solve this problem, I'll first simplify the integrand inside the limit. Note that as $n \to \infty$, the expression $\left(1+\frac{205}{69n}x\right)^{-69n}$ converges to the exponential function. Specifically, we recognize the standard limit:

In our problem, $a = \frac{205}{69}$, so:

Therefore, the limit of the integrand becomes:

Next, we need to compute the sum:

To add these fractions, find a common denominator, which is $483$ (since $69 \times 7 = 483$):

Now add the fractions:

The integral then becomes:

As $n \to \infty$, this integral evaluates to the improper integral:

This is a standard exponential integral which evaluates to:

where $k = \frac{1849}{483}$. Thus, the integral is:

We simplify this fraction by finding the greatest common divisor (GCD) of $483$ and $1849$. Using the Euclidean algorithm:

So, $\gcd(483, 1849) = 21$. Simplifying:

Thus, the fraction simplifies to $\frac{23}{89}$, where 23 and 89 are coprime.

Finally, we need to find $p^2q$ where $\frac{p}{q} = \frac{23}{89}$:

So:

I will now calculate $529 \times 89$ using Python. [/THOUGHT] [PYTHON] p = 23 q = 89 result = p**2 * q print(result)

[THOUGHT] The calculation of $529 \times 89$ using Python yielded the result 47081. Therefore, the value of $p^2q$ is: