A) \[2\,e\]
B) \[3\,e\]
C) \[(0.5)-\frac{{{(0.5)}^{2}}}{2}+\frac{{{(0.5)}^{3}}}{3}-\frac{{{(0.5)}^{4}}}{4}+....\]
D) \[5\,e\]
Correct Answer: D
Solution :
\[S=1+\frac{{{2}^{2}}}{1\ !}+\frac{{{3}^{2}}}{2\ !}+\frac{{{4}^{2}}}{3\ !}+.......+\frac{{{n}^{2}}}{(n-1)\ !}+.......\infty \] Here \[{{T}_{n}}=\frac{{{n}^{2}}}{(n-1)\ !}=\frac{(n-1)(n-2)}{(n-1)\ !}+\frac{3(n-1)+1}{(n-1)\ !}\] \[=\frac{1}{(n-3)\ !}+\frac{3}{(n-2)\ !}+\frac{1}{(n-1)\ !}\] \[\Rightarrow S=\Sigma {{T}_{n}}=e+3e+e=5e\].
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