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JEE Main & Advanced Mathematics Sequence & Series Question Bank Exponential series

  • question_answer
    In  the expansion of \[({{e}^{x}}-1)\,({{e}^{-x}}+1)\], the coefficient of \[{{x}^{3}}\] is

    A) 0

    B) 1/3

    C) 2/3

    D) 1/6

    Correct Answer: B

    Solution :

    \[({{e}^{x}}-1)({{e}^{-x}}+1)=({{e}^{x}}-1)\left( \frac{1+{{e}^{x}}}{{{e}^{x}}} \right)\]                           \[=\frac{{{e}^{2x}}-1}{{{e}^{x}}}={{e}^{x}}-{{e}^{-x}}=2\left\{ x+\frac{{{x}^{3}}}{3!}+.... \right\}\] \ The coefficient of\[{{x}^{3}}=2.\frac{1}{3!}=\frac{1}{3}\].


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