The coefficient of x2n-1in 1- 1-x 1-x2 1-x4 - 1-x2n when expanded in asscending power of x where - x - - 1 is

1/(1+x)(1+x^2)(1+x^4)...(1+x^2^n) =1 x/(1 x)(1+x)(1+x^2)....(1+x^2^n)=1 x/1 x^2^n+1 =(1 x) ( 1 x^2^n+1 )^ 1 =(1 x)(1+x^2^n+1)^ 1 =(1 x) ...

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Sum of Product of Binomial Coefficients

The coefficie...

Question

The coefficient of $x^{2^{n + 1}}$ in $\frac{1}{(1 + x) (1 + x^{2}) (1 + x^{4}) . . . (1 + x^{2^{n}})}$ (when expanded in asscending power of x) where | x | < 1 is

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x^{2^{n + 1}}

\frac{1}{(1 + x) (1 + x^{2}) (1 + x^{4}) . . . . (1 + x^{2^{n}})}

| x | < 1

| x | < 1

lim x \to \infty (1 + x) (1 + x^{2}) (1 + x^{4}) + . . . (1 + x^{2 n}) =

lim n \to \infty (1 + x) (1 + x^{2}) (1 + x^{4}) . . . (1 + x^{2 n}), | x | < 1

y = (1 + x) (1 + x^{2}) (1 + x^{4}) . . . (1 + x^{2^{n}})

\frac{d y}{d x}

x = 0

1

y = \frac{1 - x^{2^{n + 1}}}{1 - x}

| x | < 1

lim n \to \infty {(1 + x) (1 + x^{2}) (1 + x^{4}) . . . . . (1 + x^{2 n})}

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