InterviewSolution
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InterviewSolution
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Statement 1: If `log(1-x+x^(2))=a_(1)x+a_(2)x^(32)+a_(3)x^(3)+..` then `a_(3)+a_(6)+a_(9)+..=2/3log_(e)2` Statement 2: `1-1/2+1/3-11/4+1/5-1/6+..=log_(e)2`A. 1B. 2C. 3D. 4 |
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Answer» Answer: We have `log(1-x+x^(2))=a_(1)x+a_(2)x^(2)+a_(3)x^(3)+...` `rarr log(1-x+x^(2))=x(a_(1)+a_(4)x^(3)+a_(7)x^(6)+..)` `+x^(2)(a_(2)+a_(5)x^(3)+..)+(a_(3)x^(3)+a_(6)x^(6)+…)` Replacing x by 1 w and `w^(2)` respectively we get `0=e_(1)+E_(2)+E_(3)` `log(-2w)=wE_(1)+w^(2)E_(2)+E_(3)` `log(-2w^(2))=w^(2)E_(1)+wE_(2)+E_(3)` When `E_(1)=a_(1)+a_(4)+a_(7)++...` `E_(2)=a_(2)+a_(5)+..` `E_(3)=A_(3)+a_(6)+..` Adding i,ii and iii we get `log4=3 E_(3)rarrE_(3)=2/3log_(e)2` Clearly statement 2 is also true |
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