The coefficient of $x^8$ in the expansion of $\left(1 + \frac{x}{2} + \frac{x^2}{4} + \frac{x^3}{8} + \ldots\right)^{10}$ is
Step-by-Step Solution
Key Concept: Finding coefficients in products of polynomials requires summing products of coefficients from each factor that yield the desired power.
The coefficient of $x^8$ is found by multiplying corresponding terms from both factors that sum to degree 8. This equals $\frac{1}{2^8}(C_0^5 + C_1^5 + C_2^5 + C_3^5 + C_4^5) = \frac{1}{256}(C_0^5 + C_1^5 + C_2^5 + C_3^5 + C_4^5) = \frac{35}{256}$.
Correct Answer: 35