Question

Let \( S \) denote the infinite sum: \[ S = 2 + 5x + 9x^2 + 14x^3 + 20x^4 + \ldots \quad \text{where } |x|<1 \] and the coefficient of \( x^n \) is \( \frac{1}{2}n(n+3) \). Then \( S \) equals:

• A. \( \frac{2 - x}{(1 + x)^3} \)
• B. \( \frac{2 - x}{(1 - x)^3} \)
• C. \( \frac{2x}{(1 - x)^3} \)
• D. \( \frac{2 + x}{(1 + x)^3} \)

Hint:

Break complicated coefficient patterns into known power series identities like \( \sum nx^n \), \( \sum n^2x^n \) to derive closed-form.

Answer 1

The Correct Option is B

Given: \( \text{Coefficient of } x^n = \frac{1}{2}n(n+3) \Rightarrow S = \sum_{n=0}^{\infty} \frac{1}{2}n(n+3)x^n \) Break this into standard sums: \[ S = \frac{1}{2} \sum_{n=0}^{\infty} n(n+3)x^n = \frac{1}{2} \left( \sum n^2 x^n + 3 \sum n x^n \right) \] Now use known series: \[ \sum_{n=1}^{\infty} n x^n = \frac{x}{(1 - x)^2}, \quad \sum_{n=1}^{\infty} n^2 x^n = \frac{x(x+1)}{(1 - x)^3} \] Therefore: \[ S = \frac{1}{2} \left( \frac{x(x+1)}{(1 - x)^3} + 3 \cdot \frac{x}{(1 - x)^2} \right) = \frac{2 - x}{(1 - x)^3} \] \[ {\frac{2 - x}{(1 - x)^3}} \]