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Question: Coefficient of \(x^{2^{m + 1}}\)in the expansion of E=\(\frac{1}{(1 + x)(1 + x^{2})(1 + x^{4})(1 + ...

Coefficient of x2m+1x^{2^{m + 1}}in the expansion of

E=1(1+x)(1+x2)(1+x4)(1+x8)......(1+x2m)\frac{1}{(1 + x)(1 + x^{2})(1 + x^{4})(1 + x^{8})......(1 + x^{2^{m}})} (|x|< 1) is –

A

3

B

2

C

1

D

0

Answer

1

Explanation

Solution

E = 1–x1–x2\frac{1–x}{1–x^{2}} ;

E = (1 – x) (1–x2m+1)–1\left( 1–x^{2^{m + 1}} \right)^{–1}

E = (1 – x) {1+x2m+1+(x2m+1)......}\{ 1 + x^{2^{m + 1}} + (x^{2^{m + 1}})......\}

Coeffi. x2m+1x^{2^{m + 1}} = 1 ........(i)