JEE Main & Advanced Mathematics Other Series Question Bank Critical Thinking

  • question_answer The sum of the series \[1+\frac{1}{4.2\,!}+\frac{1}{16.4\,!}+\frac{1}{64.6\,!}+.....\] and inf. is [AIEEE 2005]

    A) \[\frac{e-1}{2\sqrt{e}}\]

    B) \[\frac{e+1}{2\sqrt{e}}\]

    C) \[\frac{e-1}{\sqrt{e}}\]

    D) \[\frac{e+1}{\sqrt{e}}\]

    Correct Answer: B

    Solution :

     \[\frac{{{e}^{x}}+{{e}^{-x}}}{2}=1+\frac{{{x}^{2}}}{2\,!}+\frac{{{x}^{4}}}{4\,!}+\frac{{{x}^{6}}}{6\,!}+....\infty \] Putting\[x=\frac{1}{2}\], we get \[1+\frac{1}{4\,.\,2\,!}+\frac{1}{16.4\,!}+\frac{1}{64.6!}+...\infty \]                 \[=\frac{{{e}^{1/2}}+{{e}^{-1/2}}}{2}=\frac{e+1}{2\sqrt{e}}\].

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