• # question_answer The sum of the series $1+\frac{1}{4.2\,!}+\frac{1}{16.4\,!}+\frac{1}{64.6\,!}+.....$ and inf. is [AIEEE 2005] A) $\frac{e-1}{2\sqrt{e}}$ B) $\frac{e+1}{2\sqrt{e}}$ C) $\frac{e-1}{\sqrt{e}}$ D) $\frac{e+1}{\sqrt{e}}$

$\frac{{{e}^{x}}+{{e}^{-x}}}{2}=1+\frac{{{x}^{2}}}{2\,!}+\frac{{{x}^{4}}}{4\,!}+\frac{{{x}^{6}}}{6\,!}+....\infty$ Putting$x=\frac{1}{2}$, we get $1+\frac{1}{4\,.\,2\,!}+\frac{1}{16.4\,!}+\frac{1}{64.6!}+...\infty$                 $=\frac{{{e}^{1/2}}+{{e}^{-1/2}}}{2}=\frac{e+1}{2\sqrt{e}}$.