[THOUGHT] As $n \to \infty$, the summation in $f_n(x)$ converges to a geometric series. Recognize that the partial sum $\sum_{k=1}^{n^2} (\frac{x}{2})^{k-1} \approx \frac{1}{1 - \frac{x}{2}}$. Thus, $f_n(x) \to 123456 \cdot (1 - \frac{x}{2})^2$. Compute $\int_0^2 123456 \cdot (1 - \frac{x}{2})^2 dx$.

Expand $(1 - \frac{x}{2})^2$, integrate term-by-term, and multiply by $123456$.

Final computation yields $123456 \times \frac{2}{3}$, resulting in $82208$? [/THOUGHT] [PYTHON] import sympy as sp

Define variable and integral

x = sp.symbols('x')

Compute the simplified integrand

an = 123456 * (1 - x/2)**2 integral_value = sp.integrate(an, (x, 0, 2))

Print the result as an integer

print(int(integral_value)) [/PYTHON][THOUGHT] As $n \to \infty$, the summation in $f_n(x)$ converges to a geometric series. Recognize that the partial sum $\sum_{k=1}^{n^2} (\frac{x}{2})^{k-1} \approx \frac{1}{1 - \frac{x}{2}}$. Thus, $f_n(x) \to 123456 \cdot (1 - \frac{x}{2})^2$. Compute $\int_0^2 123456 \cdot (1 - \frac{x}{2})^2 dx$.

Expand $(1 - \frac{x}{2})^2$, integrate term-by-term, and multiply by $123456$.

Final computation yields $123456 \times \frac{2}{3}$, resulting in $82208$? [/THOUGHT] [PYTHON] import sympy as sp

Define variable and integral

x = sp.symbols('x')

Compute the simplified integrand

an = 123456 * (1 - x/2)**2 integral_value = sp.integrate(an, (x, 0, 2))

Print the result as an integer

print(int(integral_value)) [/PYTHON]

[THOUGHT] Python confirmed the expected result through accurate symbolic integration. The correct value after simplifying and computing the definite integral is \boxed{82304}. [/THOUGHT] \boxed{82304}[THOUGHT] Python confirmed the expected result through accurate symbolic integration. The correct value after simplifying and computing the definite integral is \boxed{82304}. [/THOUGHT] \boxed{82304}