(B) We have,$(1 + x + x^2 + ...)^{-n} = [(1 - x)^{-1}]^{-n} = (1 - x)^n$Using the binomial expansion for $(1 - x)^n$,we get:$(1 - x)^n = \sum_{k=0}^{n

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(B) We have,$(1 + x + x^2 + ...)^{-n} = [(1 - x)^{-1}]^{-n} = (1 - x)^n$Using the binomial expansion for $(1 - x)^n$,we get:$(1 - x)^n = \sum_{k=0}^{n} \binom{n}{k} (-x)^k = \binom{n}{0} - \binom{n}{1}x + \binom{n}{2}x^2 + ... + (-1)^n \binom{n}{n} x^n$The coefficient of $x^n$ is $(-1)^n \binom{n}{n} = (-1)^n \times 1 = (-1)^n$.

Class 11 Mathematics Binomial Theorem Expansion of binomial theorem

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

A
$1$
B
$(-1)^n$
C
$n$
D
$n+1$

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Class 11

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Chapter

Binomial Theorem

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Expansion of binomial theorem

The number of terms in the expansion of $(a + b + c)^n$ will be

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The sum of all the coefficients in the binomial expansion of $(x^2 + x - 3)^{319}$ is

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If $r, k, p \in W,$ then $\sum\limits_{r + k + p = 10} {{}^{30}{C_r} \cdot {}^{20}{C_k} \cdot {}^{10}{C_p}} $ is equal to -

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The sum of the coefficients of the terms of degree $m$ in the expansion of $(1 + x)^n(1 + y)^n(1 + z)^n$ is:

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$2{C_0} + \frac{2^2}{2}{C_1} + \frac{2^3}{3}{C_2} + \dots + \frac{2^{11}}{11}{C_{10}} = \dots$

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

Similar Questions

The number of terms in the expansion of $(a + b + c)^n$ will be

The sum of all the coefficients in the binomial expansion of $(x^2 + x - 3)^{319}$ is

If $r, k, p \in W,$ then $\sum\limits_{r + k + p = 10} {{}^{30}{C_r} \cdot {}^{20}{C_k} \cdot {}^{10}{C_p}} $ is equal to -

The sum of the coefficients of the terms of degree $m$ in the expansion of $(1 + x)^n(1 + y)^n(1 + z)^n$ is:

$2{C_0} + \frac{2^2}{2}{C_1} + \frac{2^3}{3}{C_2} + \dots + \frac{2^{11}}{11}{C_{10}} = \dots$

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