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Coefficient of x^(2^(m+1)) in the expans...

Coefficient of `x^(2^(m+1))` in the expansion of `1/((1+x)(1+x^2)(1+x^4)(1+x^8).....(1+x^(2^m)))(|x|<1)` is

A

`0`

B

`1`

C

`2^(m)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
`(b)` `P=(1)/((1+x)(1+x^(2))(1+x^(4))(1+x^(8))....(1+x^(2^(m))))`
Multiplying the numerator and denominator by `1-x`, we have given expression
`=(1-x)/((1-x)(1+x)(1+x^(2))(1+x^(4))....(1+x^(2^(m))))`
`=(1-x)/((1-x^(2))(1+x^(2))(1+x^(4))....(1+x^(2^(m))))`
..........
...........
`=(1-x)/(1-x^(2^(m+1)))=(1-x)(1-x^(2^(m+1)))^(-1)`
`=(1-x)(1+x^(2^(m+1))+x^(2^(m+2))+...)`
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