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Question

The coefficient of x2n+1in1(1+x)(1+x2)(1+x4)...(1+x2n) (when expanded in asscending power of x) where | x | < 1 is

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Solution

1(1+x)(1+x2)(1+x4)...(1+x2n)
=1x(1x)(1+x)(1+x2)....(1+x2n)=1x1x2n+1
=(1x)(1x2n+1)1
=(1x)(1+x2n+1)1
=(1x)(1+x2n+1)+(x2n+1)2+....)
So, the coefficient of x2n+1=1

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