JEE Main & Advanced Mathematics Definite Integration Question Bank Summation of series by Definite Integration, Gamma function, Leibnitz's rule

  • question_answer
    \[\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{r=1}^{n}{\frac{1}{n}{{e}^{\frac{r}{n}}}}\]is         [AIEEE 2004]

    A)                 \[e+1\]

    B)                 \[e-1\]

    C)                 \[1-e\] 

    D)                 \[e\]

    Correct Answer: B

    Solution :

                    \[\underset{n\to \infty }{\mathop{\lim }}\,\sum\limits_{r=1}^{n}{\frac{1}{n}{{e}^{\frac{r}{n}}}=\int_{0}^{1}{{{e}^{x}}dx=[{{e}^{x}}]_{0}^{1}=e-1}}\].


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