Question

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

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

Hint:

For infinite series with polynomial coefficients, use known series sum formulas for fast results.

Answer 1

The Correct Option is A

The series \( 2 + 5x + 9x^2 + 14x^3 + 20x^4 + \dots \) is a standard series whose general form is \( a_n x^n \) with the coefficient of \( x^n \) given by \( a_n = \frac{1}{2} n(n + 3) \). The sum of the series is known to follow a standard formula for such series: \[ S = \frac{2 - x}{(1 - x)^3}. \] Thus, the Correct Answer is \( \frac{2 - x}{(1 - x)^3} \).