A) \[{{e}^{4}}\]
B) \[\frac{{{e}^{2}}-1}{{{e}^{2}}}\]
C) \[\frac{{{e}^{4}}-1}{4\,{{e}^{2}}}\]
D) \[\frac{{{e}^{4}}+1}{4\,{{e}^{2}}}\]
Correct Answer: C
Solution :
\[\left( 1+\frac{1}{2\ !}+\frac{1}{4\ !}+..... \right)\,\left( 1+\frac{1}{3\ !}+\frac{1}{5\ !}+....... \right)\] \[=\frac{e+{{e}^{-1}}}{2}.\frac{e-{{e}^{-1}}}{2}=\frac{{{e}^{2}}-{{e}^{-2}}}{4}=\frac{{{e}^{4}}-1}{4{{e}^{2}}}\].
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