The coefficient of xn in the expansion of (1 - 2x + 3x2 - 4x3 + ... to ∞)-n is

Question

The coefficient of xn in the expansion of (1 - 2x + 3x2 - 4x3 + ... to ∞)-n is (A) ((2n)!/n!(n-1)!) (B) ((2n)!/[(n-1)!]2) (C) ((2n)!/(n!)2) (D) None of these

Tardigrade Admin · Accepted Answer

We have, (1 - 2x + 3x2 - 4x3 + ... to ∞)-n = [(1+x)-2]-n = (1+x)2n ∴ Coefficient of xn = 2nCn = ((2n)!/(n!)2).

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

$\frac{( 2 n ) !}{n ! ( n - 1 ) !}$

$\frac{( 2 n ) !}{[ ( n - 1 ) ! ] ^{2}}$

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

None of these

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

Q. The coefficient of xn in the expansion of (1−2x+3x2−4x3+... to ∞)−n is

n!(n−1)!(2n)!​

[(n−1)!]2(2n)!​

(n!)2(2n)!​