Problem Analysis #5

[THOUGHT] For each fixed $x\in[0,2]$, the factor $\left(\frac{x}{2}\right)^{k-1}\left(1-\frac{x}{2}\right)^3$ is a geometric term in $k$. Since $0\le \frac{x}{2}\le 1$, the infinite geometric series

converges to $\left(1-\frac{x}{2}\right)^3 \cdot \frac{1}{1-\frac{x}{2}} = \left(1-\frac{x}{2}\right)^2$ for $x\neq 2$, and at $x=2$ the sum is $n^2\cdot 123456\cdot 0^3=0$. Hence $f_n(x)\to 123456\left(1-\frac{x}{2}\right)^2$ pointwise on $[0,2)$, and the limit function is integrable. By the Dominated Convergence Theorem, we can pass the limit inside the integral:

Compute this integral exactly using Python. [/THOUGHT] [PYTHON] from sympy import integrate, simplify, lambdify, Symbol

x = Symbol('x') I = 123456 * integrate((1 - x/2)**2, (x, 0, 2)) print(I) [/PYTHON]

[THOUGHT] Using the Python output as ground truth, the value is $\boxed{82304}$.