JEE Main & Advanced Mathematics Other Series Question Bank Critical Thinking

  • question_answer 2) \[\frac{1}{1\,!}+\frac{4}{2\,!}+\frac{7}{3\,!}+\frac{10}{4\,!}+.....\infty =\]

    A) \[e+4\]

    B) \[2+e\]

    C) \[3+e\]

    D) \[e\]

    Correct Answer: B

    Solution :

    \[S=\frac{1}{1\ !}+\frac{4}{2\ !}+\frac{7}{3\ !}+\frac{10}{4\ !}+......+\frac{3n-2}{n\ !}+.......\infty \] Here \[{{T}_{n}}=\frac{3}{(n-1)\ !}-\frac{2}{n\ !}\] \[\Rightarrow S=\sum\limits_{n=1}^{\infty }{{{T}_{n}}=3\sum\limits_{n=1}^{\infty }{\frac{1}{(n-1)\ !}-2\sum\limits_{n=1}^{\infty }{\frac{1}{n\ !}}}}\]        \[=3e-2(e-1)=e+2\].

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