A) \[e(e+1)\]
B) \[e\,(1-e)\]
C) \[3e-1\]
D) \[3e\]
E) (e) \[e\,(e-1)\]
Correct Answer: E
Solution :
(e) \[1+\frac{3}{2\,!}+\frac{7}{3\,!}+\frac{15}{4\,!}+....\] \[=\,(1-1)+\left( \frac{2}{1!}-\frac{1}{1!} \right)+\left( \frac{{{2}^{2}}}{2\,!}-\frac{{{1}^{2}}}{2\,!} \right)+\left( \frac{{{2}^{3}}}{3\,!}-\frac{{{1}^{3}}}{3\,!} \right)+...\] \[=\left( 1+\frac{2}{1!}+\frac{{{2}^{2}}}{2!}+\frac{{{2}^{3}}}{3!}+.... \right)\]\[-\left( 1+\frac{1}{1!}+\frac{{{1}^{2}}}{2!}+\frac{{{1}^{3}}}{3!}+... \right)\] \[={{e}^{2}}-e\] = \[e(e-1)\].
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