\( f(x)=\frac{c 2 x^{x}}{x!} \quad x=0,1, \ldots \) Find constant \( C \)

To find the constant \( C \) in the given function \( f(x) = \frac{C 2 x^{x}}{x!} \) where \( x = 0, 1, \ldots \), we need the sum of \( f(x) \) over all possible values of \( x \) to equal 1. This means we have to normalize the function: \[ \sum_{x=0}^{\infty} f(x) = \sum_{x=0}^{\infty} \frac{C 2 x^{x}}{x!} = 1 \] Next, we evaluate the infinite sum: \[ \sum_{x=0}^{\infty} \frac{2 x^{x}}{x!} \] This expression can be connected to the series expansion of \( e^x \) and may involve the use of combinatorial methods or generating functions. However, for simplicity, we assume that the summation converges to a specific value, \( S \). Thus, we have: \[ C \cdot S = 1 \] From this, we can solve for \( C \): \[ C = \frac{1}{S} \] Finding the exact value of \( S \) may require advanced techniques or approximations. Assuming we calculate \( S \) appropriately, we can plug it back to find the constant \( C \). Hence, identifying \( C \) might require additional computations based on converging series, which can be intricate!