Question

When \( |x|>3 \), the coefficient of \( \frac{1}{x^n} \) in the expansion of \( x^{3/2} (3+x)^{1/2} \) is

• A. \( (-1)^n \frac{1.3.5 \dots (2n-1)}{2^n n!} 3^n \)
• B. \( (-1)^{n+1} \frac{1.3.5 \dots (2n+1)}{2^{n+2} (n+2)!} 3^{n+2} \)
• C. \( (-1)^{n+1} \frac{1.3.5 \dots (2n-1)}{2^n n!} 3^{n+1} \)
• D. \( (-1)^{n+1} \frac{1.3.5 \dots (2n+1)}{2^{n+3} (n+2)!} 3^{n+1} \)

Hint:

For binomial expansions of \( (a+x)^n \) where \( n \) is negative or fractional, always factor out the term with the larger magnitude to ensure the expansion converges (ratio \(<1 \)).

Answer 1

The Correct Option is B