Question

The coefficient of $x^n$ in the expansion of ($1 - 2x + 3x^{2} - 4x^{3} + ...$ to $\infty$)$^{-n}$ is

• A. $\frac{\left(2n\right)!}{n!\left(n-1\right)!}$
• B. $\frac{\left(2n\right)!}{\left[\left(n-1\right)!\right]^{2}}$
• C. $\frac{\left(2n\right)!}{\left(n!\right)^{2}}$
• D. None of these

Answer 1

Answer: (C)

$\frac{\left(2n\right)!}{\left(n!\right)^{2}}$

Answer 2

We have, ($1 - 2x + 3x^{2} - 4x^{3} + ...$ to $\infty$)$^{-n}$ $= \left[\left(1+x\right)^{-2}\right]^{-n} = \left(1+x\right)^{2n}$ $\therefore$ Coefficient of $x^{n} =\,^{2n}C_{n} = \frac{\left(2n\right)!}{\left(n!\right)^{2}}$.